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Theorem sltres 32146
Description: If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
sltres ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → 𝐴 <s 𝐵))

Proof of Theorem sltres
Dummy variables 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noreson 32144 . . . . . . 7 ((𝐴 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
213adant2 1124 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
3 noreson 32144 . . . . . . 7 ((𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
433adant1 1123 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
5 sltintdifex 32145 . . . . . . 7 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V))
6 onintrab 7147 . . . . . . 7 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V ↔ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
75, 6syl6ib 241 . . . . . 6 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
82, 4, 7syl2anc 565 . . . . 5 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
98imp 393 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
10 simpl3 1230 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝑋 ∈ On)
11 sltval2 32140 . . . . . . . . . . . 12 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) ↔ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
122, 4, 11syl2anc 565 . . . . . . . . . . 11 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) ↔ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
13 fvex 6342 . . . . . . . . . . . . 13 ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
14 fvex 6342 . . . . . . . . . . . . 13 ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
1513, 14brtp 31971 . . . . . . . . . . . 12 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
16 1n0 7728 . . . . . . . . . . . . . . . . . 18 1𝑜 ≠ ∅
1716neii 2944 . . . . . . . . . . . . . . . . 17 ¬ 1𝑜 = ∅
18 eqeq1 2774 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 1𝑜 = ∅))
1917, 18mtbiri 316 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
20 ndmfv 6359 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
2119, 20nsyl2 144 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2221adantr 466 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2322orcd 853 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
2421adantr 466 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2524orcd 853 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
26 2on 7721 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 ∈ On
2726elexi 3362 . . . . . . . . . . . . . . . . . . . 20 2𝑜 ∈ V
2827prid2 4432 . . . . . . . . . . . . . . . . . . 19 2𝑜 ∈ {1𝑜, 2𝑜}
2928nosgnn0i 32143 . . . . . . . . . . . . . . . . . 18 ∅ ≠ 2𝑜
3029neii 2944 . . . . . . . . . . . . . . . . 17 ¬ ∅ = 2𝑜
31 eqeq1 2774 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 2𝑜 = ∅))
32 eqcom 2777 . . . . . . . . . . . . . . . . . 18 (2𝑜 = ∅ ↔ ∅ = 2𝑜)
3331, 32syl6bb 276 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ ∅ = 2𝑜))
3430, 33mtbiri 316 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
35 ndmfv 6359 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
3634, 35nsyl2 144 . . . . . . . . . . . . . . 15 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
3736adantl 467 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
3837olcd 854 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
3923, 25, 383jaoi 1538 . . . . . . . . . . . 12 (((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
4015, 39sylbi 207 . . . . . . . . . . 11 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
4112, 40syl6bi 243 . . . . . . . . . 10 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))))
4241imp 393 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
43 dmres 5560 . . . . . . . . . . . 12 dom (𝐴𝑋) = (𝑋 ∩ dom 𝐴)
4443elin2 3950 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ↔ ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴))
4544simplbi 479 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
46 dmres 5560 . . . . . . . . . . . 12 dom (𝐵𝑋) = (𝑋 ∩ dom 𝐵)
4746elin2 3950 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) ↔ ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵))
4847simplbi 479 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
4945, 48jaoi 837 . . . . . . . . 9 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) ∨ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
5042, 49syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
51 onelss 5909 . . . . . . . 8 (𝑋 ∈ On → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑋))
5210, 50, 51sylc 65 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑋)
5352sselda 3750 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦𝑋)
54 onelon 5891 . . . . . . . . 9 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
559, 54sylan 561 . . . . . . . 8 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
56 intss1 4624 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑦)
57 ontri1 5900 . . . . . . . . . . . . 13 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑦 ↔ ¬ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
5856, 57syl5ib 234 . . . . . . . . . . . 12 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
5958con2d 131 . . . . . . . . . . 11 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
609, 59sylan 561 . . . . . . . . . 10 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 ∈ On) → (𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6160impancom 439 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝑦 ∈ On → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6255, 61mpd 15 . . . . . . . 8 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})
63 fveq2 6332 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐴𝑋)‘𝑎) = ((𝐴𝑋)‘𝑦))
64 fveq2 6332 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐵𝑋)‘𝑎) = ((𝐵𝑋)‘𝑦))
6563, 64neeq12d 3003 . . . . . . . . . . 11 (𝑎 = 𝑦 → (((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎) ↔ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6665elrab 3513 . . . . . . . . . 10 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ↔ (𝑦 ∈ On ∧ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6766simplbi2 482 . . . . . . . . 9 (𝑦 ∈ On → (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) → 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6867con3d 149 . . . . . . . 8 (𝑦 ∈ On → (¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6955, 62, 68sylc 65 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦))
70 df-ne 2943 . . . . . . . 8 (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
7170con2bii 346 . . . . . . 7 (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦))
7269, 71sylibr 224 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
73 fvres 6348 . . . . . . . 8 (𝑦𝑋 → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
74 fvres 6348 . . . . . . . 8 (𝑦𝑋 → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
7573, 74eqeq12d 2785 . . . . . . 7 (𝑦𝑋 → (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
7675biimpd 219 . . . . . 6 (𝑦𝑋 → (((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦) → (𝐴𝑦) = (𝐵𝑦)))
7753, 72, 76sylc 65 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝐴𝑦) = (𝐵𝑦))
7877ralrimiva 3114 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦))
79 fvresval 31997 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8079ori 841 . . . . . . . . . . . . . 14 (¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8119, 80nsyl2 144 . . . . . . . . . . . . 13 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
8281eqcomd 2776 . . . . . . . . . . . 12 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
83 eqeq2 2781 . . . . . . . . . . . 12 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜))
8482, 83mpbid 222 . . . . . . . . . . 11 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜)
8584adantr 466 . . . . . . . . . 10 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜)
8685a1i 11 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜))
8721ad2antrl 699 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
8887, 45syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
89 nofun 32133 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → Fun (𝐵𝑋))
90 fvelrn 6495 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐵𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋))
9190ex 397 . . . . . . . . . . . . . . . . . 18 (Fun (𝐵𝑋) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋)))
9289, 91syl 17 . . . . . . . . . . . . . . . . 17 ((𝐵𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋)))
93 norn 32135 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → ran (𝐵𝑋) ⊆ {1𝑜, 2𝑜})
9493sseld 3749 . . . . . . . . . . . . . . . . 17 ((𝐵𝑋) ∈ No → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
9592, 94syld 47 . . . . . . . . . . . . . . . 16 ((𝐵𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
96 nosgnn0 32142 . . . . . . . . . . . . . . . . 17 ¬ ∅ ∈ {1𝑜, 2𝑜}
97 eleq1 2837 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
9896, 97mtbiri 316 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
9995, 98nsyli 156 . . . . . . . . . . . . . . 15 ((𝐵𝑋) ∈ No → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
1004, 99syl 17 . . . . . . . . . . . . . 14 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
101100imp 393 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
102101adantrl 687 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
10347simplbi2 482 . . . . . . . . . . . . 13 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)))
104103con3d 149 . . . . . . . . . . . 12 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → (¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵))
10588, 102, 104sylc 65 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵)
106 ndmfv 6359 . . . . . . . . . . 11 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐵 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
107105, 106syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
108107ex 397 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅))
10986, 108jcad 496 . . . . . . . 8 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)))
110 fvresval 31997 . . . . . . . . . . . . . 14 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
111110ori 841 . . . . . . . . . . . . 13 (¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
11234, 111nsyl2 144 . . . . . . . . . . . 12 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
113112eqcomd 2776 . . . . . . . . . . 11 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
114 eqeq2 2781 . . . . . . . . . . 11 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ((𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))
115113, 114mpbid 222 . . . . . . . . . 10 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)
11684, 115anim12i 592 . . . . . . . . 9 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))
117116a1i 11 . . . . . . . 8 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
11836ad2antll 700 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
119118, 48syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
120 nofun 32133 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → Fun (𝐴𝑋))
121 fvelrn 6495 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐴𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋))
122121ex 397 . . . . . . . . . . . . . . . . . 18 (Fun (𝐴𝑋) → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋)))
123120, 122syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋)))
124 norn 32135 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → ran (𝐴𝑋) ⊆ {1𝑜, 2𝑜})
125124sseld 3749 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋) ∈ No → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
126123, 125syld 47 . . . . . . . . . . . . . . . 16 ((𝐴𝑋) ∈ No → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
127 eleq1 2837 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
12896, 127mtbiri 316 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
129126, 128nsyli 156 . . . . . . . . . . . . . . 15 ((𝐴𝑋) ∈ No → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)))
1302, 129syl 17 . . . . . . . . . . . . . 14 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)))
131130imp 393 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
132131adantrr 688 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
13344simplbi2 482 . . . . . . . . . . . . 13 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)))
134133con3d 149 . . . . . . . . . . . 12 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋 → (¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴))
135119, 132, 134sylc 65 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴)
136135ex 397 . . . . . . . . . 10 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴))
137 ndmfv 6359 . . . . . . . . . 10 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom 𝐴 → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
138136, 137syl6 35 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅))
139115adantl 467 . . . . . . . . . 10 ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)
140139a1i 11 . . . . . . . . 9 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))
141138, 140jcad 496 . . . . . . . 8 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) → ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
142109, 117, 1413orim123d 1554 . . . . . . 7 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)) → (((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜))))
143 fvex 6342 . . . . . . . 8 (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
144 fvex 6342 . . . . . . . 8 (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
145143, 144brtp 31971 . . . . . . 7 ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ↔ (((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜) ∨ ((𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜)))
146142, 15, 1453imtr4g 285 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
14712, 146sylbid 230 . . . . 5 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
148147imp 393 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
149 raleq 3286 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦)))
150 fveq2 6332 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
151 fveq2 6332 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
152150, 151breq12d 4797 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ((𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥) ↔ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)})))
153149, 152anbi12d 608 . . . . 5 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ (∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))))
154153rspcev 3458 . . . 4 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
1559, 78, 148, 154syl12anc 1473 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
156 sltval 32131 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
1571563adant3 1125 . . . 4 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
158157adantr 466 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
159155, 158mpbird 247 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝐴 <s 𝐵)
160159ex 397 1 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → 𝐴 <s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  w3o 1069  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  wss 3721  c0 4061  {cpr 4316  {ctp 4318  cop 4320   cint 4609   class class class wbr 4784  dom cdm 5249  ran crn 5250  cres 5251  Oncon0 5866  Fun wfun 6025  cfv 6031  1𝑜c1o 7705  2𝑜c2o 7706   No csur 32124   <s cslt 32125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7712  df-2o 7713  df-no 32127  df-slt 32128
This theorem is referenced by:  noresle  32177  nosupbnd1lem1  32185  nosupbnd1lem2  32186  nosupbnd1  32191  nosupbnd2lem1  32192  nosupbnd2  32193  noetalem2  32195  noetalem3  32196
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