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Theorem nodense 31967
Description: Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD). (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodense (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 No (( bday 𝑥) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑥𝑥 <s 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nodense
Dummy variables 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nodenselem6 31964 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )
2 bdayval 31926 . . . . 5 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No → ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) = dom (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
31, 2syl 17 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) = dom (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
4 dmres 5454 . . . . 5 dom (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∩ dom 𝐴)
5 nodenselem5 31963 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
6 bdayfo 31953 . . . . . . . . . . 11 bday : No onto→On
7 fof 6153 . . . . . . . . . . 11 ( bday : No onto→On → bday : No ⟶On)
86, 7ax-mp 5 . . . . . . . . . 10 bday : No ⟶On
9 0elon 5816 . . . . . . . . . 10 ∅ ∈ On
108, 9f0cli 6410 . . . . . . . . 9 ( bday 𝐴) ∈ On
1110onelssi 5874 . . . . . . . 8 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ ( bday 𝐴))
125, 11syl 17 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ ( bday 𝐴))
13 bdayval 31926 . . . . . . . 8 (𝐴 No → ( bday 𝐴) = dom 𝐴)
1413ad2antrr 762 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐴) = dom 𝐴)
1512, 14sseqtrd 3674 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ dom 𝐴)
16 df-ss 3621 . . . . . 6 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ dom 𝐴 ↔ ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∩ dom 𝐴) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
1715, 16sylib 208 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∩ dom 𝐴) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
184, 17syl5eq 2697 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → dom (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
193, 18eqtrd 2685 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
2019, 5eqeltrd 2730 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ∈ ( bday 𝐴))
21 nodenselem4 31962 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
2221adantrl 752 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
23 nodenselem8 31966 . . . . . . . . . . . . 13 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)))
2423biimpd 219 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)))
25243expia 1286 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (( bday 𝐴) = ( bday 𝐵) → (𝐴 <s 𝐵 → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜))))
2625imp32 448 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜))
2726simpld 474 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜)
28 eqid 2651 . . . . . . . . 9 ∅ = ∅
2927, 28jctir 560 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ ∅ = ∅))
30293mix1d 1256 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ ∅ = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ ∅ = 2𝑜) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ ∅ = 2𝑜)))
31 fvex 6239 . . . . . . . 8 (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
32 0ex 4823 . . . . . . . 8 ∅ ∈ V
3331, 32brtp 31765 . . . . . . 7 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}∅ ↔ (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ ∅ = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ ∅ = 2𝑜) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ ∅ = 2𝑜)))
3430, 33sylibr 224 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}∅)
3519fveq2d 6233 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
36 fvnobday 31954 . . . . . . . 8 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))) = ∅)
371, 36syl 17 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))) = ∅)
3835, 37eqtr3d 2687 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
3934, 38breqtrrd 4713 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
40 fvres 6245 . . . . . . 7 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐴𝑦))
4140eqcomd 2657 . . . . . 6 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦))
4241rgen 2951 . . . . 5 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦)
4339, 42jctil 559 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
44 raleq 3168 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (∀𝑦𝑥 (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦)))
45 fveq2 6229 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
46 fveq2 6229 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
4745, 46breq12d 4698 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥) ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
4844, 47anbi12d 747 . . . . 5 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((∀𝑦𝑥 (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥)) ↔ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))))
4948rspcev 3340 . . . 4 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥)))
5022, 43, 49syl2anc 694 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥)))
51 simpll 805 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
52 sltval 31925 . . . 4 ((𝐴 No ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No ) → (𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥))))
5351, 1, 52syl2anc 694 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥))))
5450, 53mpbird 247 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
5541adantl 481 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦))
56 nodenselem7 31965 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑦) = (𝐵𝑦)))
5756imp 444 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝑦) = (𝐵𝑦))
5855, 57eqtr3d 2687 . . . . 5 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦))
5958ralrimiva 2995 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦))
6026simprd 478 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)
6160, 28jctil 559 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (∅ = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜))
62613mix3d 1258 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((∅ = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ (∅ = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜) ∨ (∅ = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)))
63 fvex 6239 . . . . . . 7 (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
6432, 63brtp 31765 . . . . . 6 (∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ ((∅ = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ (∅ = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜) ∨ (∅ = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)))
6562, 64sylibr 224 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
6638, 65eqbrtrd 4707 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
67 raleq 3168 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (∀𝑦𝑥 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ↔ ∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦)))
68 fveq2 6229 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
6946, 68breq12d 4698 . . . . . 6 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥) ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
7067, 69anbi12d 747 . . . . 5 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((∀𝑦𝑥 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))))
7170rspcev 3340 . . . 4 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (∀𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))) → ∃𝑥 ∈ On (∀𝑦𝑥 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
7222, 59, 66, 71syl12anc 1364 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ On (∀𝑦𝑥 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
73 simplr 807 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
74 sltval 31925 . . . 4 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No 𝐵 No ) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
751, 73, 74syl2anc 694 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐵𝑦) ∧ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
7672, 75mpbird 247 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵)
77 fveq2 6229 . . . . 5 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ( bday 𝑥) = ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
7877eleq1d 2715 . . . 4 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (( bday 𝑥) ∈ ( bday 𝐴) ↔ ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ∈ ( bday 𝐴)))
79 breq2 4689 . . . 4 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴 <s 𝑥𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
80 breq1 4688 . . . 4 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝑥 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵))
8178, 79, 803anbi123d 1439 . . 3 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ((( bday 𝑥) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑥𝑥 <s 𝐵) ↔ (( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ∈ ( bday 𝐴) ∧ 𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵)))
8281rspcev 3340 . 2 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No ∧ (( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ∈ ( bday 𝐴) ∧ 𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵)) → ∃𝑥 No (( bday 𝑥) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑥𝑥 <s 𝐵))
831, 20, 54, 76, 82syl13anc 1368 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 No (( bday 𝑥) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑥𝑥 <s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  cin 3606  wss 3607  c0 3948  {ctp 4214  cop 4216   cint 4507   class class class wbr 4685  dom cdm 5143  cres 5145  Oncon0 5761  wf 5922  ontowfo 5924  cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918   <s cslt 31919   bday cbday 31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922  df-bday 31923
This theorem is referenced by:  nocvxminlem  32018
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