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Theorem sltres 31799
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 31797 . . . . . . 7 ((𝐴 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
213adant2 1079 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴𝑋) ∈ No )
3 noreson 31797 . . . . . . 7 ((𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
433adant1 1078 . . . . . 6 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐵𝑋) ∈ No )
5 sltintdifex 31798 . . . . . . 7 (((𝐴𝑋) ∈ No ∧ (𝐵𝑋) ∈ No ) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ V))
6 onintrab 6998 . . . . . . 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 693 . . . . 5 ((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On))
98imp 445 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On)
10 simpl3 1065 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → 𝑋 ∈ On)
11 sltval2 31793 . . . . . . . . . . . 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 693 . . . . . . . . . . 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 6199 . . . . . . . . . . . . 13 ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
14 fvex 6199 . . . . . . . . . . . . 13 ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
1513, 14brtp 31625 . . . . . . . . . . . 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 7572 . . . . . . . . . . . . . . . . . 18 1𝑜 ≠ ∅
1716neii 2795 . . . . . . . . . . . . . . . . 17 ¬ 1𝑜 = ∅
18 eqeq1 2625 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 1𝑜 = ∅))
1917, 18mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
20 ndmfv 6216 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
2119, 20nsyl2 142 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
2221adantr 481 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 7565 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 ∈ On
2726elexi 3211 . . . . . . . . . . . . . . . . . . . 20 2𝑜 ∈ V
2827prid2 4296 . . . . . . . . . . . . . . . . . . 19 2𝑜 ∈ {1𝑜, 2𝑜}
2928nosgnn0i 31796 . . . . . . . . . . . . . . . . . 18 ∅ ≠ 2𝑜
3029neii 2795 . . . . . . . . . . . . . . . . 17 ¬ ∅ = 2𝑜
31 eqeq1 2625 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ ↔ 2𝑜 = ∅))
32 eqcom 2628 . . . . . . . . . . . . . . . . . 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 317 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
35 ndmfv 6216 . . . . . . . . . . . . . . . 16 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
3634, 35nsyl2 142 . . . . . . . . . . . . . . 15 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
3736adantl 482 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 1390 . . . . . . . . . . . 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 445 . . . . . . . . 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 5417 . . . . . . . . . . . 12 dom (𝐴𝑋) = (𝑋 ∩ dom 𝐴)
4443elin2 3799 . . . . . . . . . . 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 476 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
46 dmres 5417 . . . . . . . . . . . 12 dom (𝐵𝑋) = (𝑋 ∩ dom 𝐵)
4746elin2 3799 . . . . . . . . . . 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 476 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋) → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ 𝑋)
4945, 48jaoi 394 . . . . . . . . 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 5764 . . . . . . . 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 3601 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦𝑋)
54 onelon 5746 . . . . . . . . 9 (( {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ On ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
559, 54sylan 488 . . . . . . . 8 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) ∧ 𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → 𝑦 ∈ On)
56 intss1 4490 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ⊆ 𝑦)
57 ontri1 5755 . . . . . . . . . . . . 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 129 . . . . . . . . . . 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 488 . . . . . . . . . 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 456 . . . . . . . . 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 6189 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐴𝑋)‘𝑎) = ((𝐴𝑋)‘𝑦))
64 fveq2 6189 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝐵𝑋)‘𝑎) = ((𝐵𝑋)‘𝑦))
6563, 64neeq12d 2854 . . . . . . . . . . 11 (𝑎 = 𝑦 → (((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎) ↔ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6665elrab 3361 . . . . . . . . . 10 (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ↔ (𝑦 ∈ On ∧ ((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦)))
6766simplbi2 655 . . . . . . . . 9 (𝑦 ∈ On → (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) → 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
6867con3d 148 . . . . . . . 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 2794 . . . . . . . 8 (((𝐴𝑋)‘𝑦) ≠ ((𝐵𝑋)‘𝑦) ↔ ¬ ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
7170con2bii 347 . . . . . . 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 6205 . . . . . . . 8 (𝑦𝑋 → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
74 fvres 6205 . . . . . . . 8 (𝑦𝑋 → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
7573, 74eqeq12d 2636 . . . . . . 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 2965 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦))
79 fvresval 31651 . . . . . . . . . . . . . . 15 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8079ori 390 . . . . . . . . . . . . . 14 (¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
8119, 80nsyl2 142 . . . . . . . . . . . . 13 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 1𝑜 → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
8281eqcomd 2627 . . . . . . . . . . . 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 2632 . . . . . . . . . . . 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 481 . . . . . . . . . 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 764 . . . . . . . . . . . . 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 31786 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → Fun (𝐵𝑋))
90 fvelrn 6350 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐵𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋)) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐵𝑋))
9190ex 450 . . . . . . . . . . . . . . . . . 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 31788 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑋) ∈ No → ran (𝐵𝑋) ⊆ {1𝑜, 2𝑜})
9493sseld 3600 . . . . . . . . . . . . . . . . 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 31795 . . . . . . . . . . . . . . . . 17 ¬ ∅ ∈ {1𝑜, 2𝑜}
97 eleq1 2688 . . . . . . . . . . . . . . . . 17 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
9896, 97mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
9995, 98nsyli 155 . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐵𝑋))
102101adantrl 752 . . . . . . . . . . . 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 655 . . . . . . . . . . . . 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 148 . . . . . . . . . . . 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 6216 . . . . . . . . . . 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 450 . . . . . . . . 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 555 . . . . . . . 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 31651 . . . . . . . . . . . . . 14 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∨ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
111110ori 390 . . . . . . . . . . . . 13 (¬ ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅)
11234, 111nsyl2 142 . . . . . . . . . . . 12 (((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = 2𝑜 → ((𝐵𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
113112eqcomd 2627 . . . . . . . . . . 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 2632 . . . . . . . . . . 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 590 . . . . . . . . 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 765 . . . . . . . . . . . . 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 31786 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → Fun (𝐴𝑋))
121 fvelrn 6350 . . . . . . . . . . . . . . . . . . 19 ((Fun (𝐴𝑋) ∧ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋)) → ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ ran (𝐴𝑋))
122121ex 450 . . . . . . . . . . . . . . . . . 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 31788 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋) ∈ No → ran (𝐴𝑋) ⊆ {1𝑜, 2𝑜})
125124sseld 3600 . . . . . . . . . . . . . . . . 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 2688 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
12896, 127mtbiri 317 . . . . . . . . . . . . . . . 16 (((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅ → ¬ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
129126, 128nsyli 155 . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋)‘ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) = ∅) → ¬ {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} ∈ dom (𝐴𝑋))
132131adantrr 753 . . . . . . . . . . . 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 655 . . . . . . . . . . . . 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 148 . . . . . . . . . . . 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 450 . . . . . . . . . 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 6216 . . . . . . . . . 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 482 . . . . . . . . . 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 555 . . . . . . . 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 1406 . . . . . . 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 6199 . . . . . . . 8 (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
144 fvex 6199 . . . . . . . 8 (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}) ∈ V
145143, 144brtp 31625 . . . . . . 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 445 . . . 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 3136 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} (𝐴𝑦) = (𝐵𝑦)))
150 fveq2 6189 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
151 fveq2 6189 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ ((𝐴𝑋)‘𝑎) ≠ ((𝐵𝑋)‘𝑎)}))
152150, 151breq12d 4664 . . . . . 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 747 . . . . 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 3307 . . . 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 1323 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ (𝐴𝑋) <s (𝐵𝑋)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
156 sltval 31784 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
1571563adant3 1080 . . . 4 ((𝐴 No 𝐵 No 𝑋 ∈ On) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
158157adantr 481 . . 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 450 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  wo 383  wa 384  w3o 1036  w3a 1037   = wceq 1482  wcel 1989  wne 2793  wral 2911  wrex 2912  {crab 2915  Vcvv 3198  wss 3572  c0 3913  {cpr 4177  {ctp 4179  cop 4181   cint 4473   class class class wbr 4651  dom cdm 5112  ran crn 5113  cres 5114  Oncon0 5721  Fun wfun 5880  cfv 5886  1𝑜c1o 7550  2𝑜c2o 7551   No csur 31777   <s cslt 31778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-ord 5724  df-on 5725  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-1o 7557  df-2o 7558  df-no 31780  df-slt 31781
This theorem is referenced by:  noresle  31830  nosupbnd1lem1  31838  nosupbnd1lem2  31839  nosupbnd1  31844  nosupbnd2lem1  31845  nosupbnd2  31846  noetalem2  31848  noetalem3  31849
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