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Theorem nodense 33099
 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 33096 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )
2 bdayval 33058 . . . . 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 5874 . . . . 5 dom (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∩ dom 𝐴)
5 nodenselem5 33095 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
6 bdayfo 33085 . . . . . . . . . . 11 bday : No onto→On
7 fof 6589 . . . . . . . . . . 11 ( bday : No onto→On → bday : No ⟶On)
86, 7ax-mp 5 . . . . . . . . . 10 bday : No ⟶On
9 0elon 6243 . . . . . . . . . 10 ∅ ∈ On
108, 9f0cli 6862 . . . . . . . . 9 ( bday 𝐴) ∈ On
1110onelssi 6298 . . . . . . . 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 33058 . . . . . . . 8 (𝐴 No → ( bday 𝐴) = dom 𝐴)
1413ad2antrr 722 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐴) = dom 𝐴)
1512, 14sseqtrd 4011 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ dom 𝐴)
16 df-ss 3956 . . . . . 6 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ dom 𝐴 ↔ ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∩ dom 𝐴) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
1715, 16sylib 219 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∩ dom 𝐴) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
184, 17syl5eq 2873 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → dom (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
193, 18eqtrd 2861 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
2019, 5eqeltrd 2918 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ∈ ( bday 𝐴))
21 nodenselem4 33094 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
2221adantrl 712 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
23 nodenselem8 33098 . . . . . . . . . . . . 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 230 . . . . . . . . . . . 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 1115 . . . . . . . . . . 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 419 . . . . . . . . . 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 495 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o)
28 eqid 2826 . . . . . . . . 9 ∅ = ∅
2927, 28jctir 521 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ ∅ = ∅))
30293mix1d 1330 . . . . . . 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 6682 . . . . . . . 8 (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
32 0ex 5208 . . . . . . . 8 ∅ ∈ V
3331, 32brtp 32888 . . . . . . 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 235 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅)
3519fveq2d 6673 . . . . . . 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 33086 . . . . . . . 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 2863 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
3934, 38breqtrrd 5091 . . . . 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 6688 . . . . . . 7 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦) = (𝐴𝑦))
4140eqcomd 2832 . . . . . 6 (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦))
4241rgen 3153 . . . . 5 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} (𝐴𝑦) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑦)
4339, 42jctil 520 . . . 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 3411 . . . . . 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 6669 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑥) = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
46 fveq2 6669 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘𝑥) = ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})‘ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
4745, 46breq12d 5076 . . . . . 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 630 . . . . 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 3627 . . . 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 584 . . 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 763 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
52 sltval 33057 . . . 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 584 . . 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 258 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
5541adantl 482 . . . . . 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 33097 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝑦) = (𝐵𝑦)))
5756imp 407 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝑦 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝑦) = (𝐵𝑦))
5855, 57eqtr3d 2863 . . . . 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 3187 . . . 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 496 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)
6160, 28jctil 520 . . . . . . 7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (∅ = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o))
62613mix3d 1332 . . . . . 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 6682 . . . . . . 7 (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
6432, 63brtp 32888 . . . . . 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 235 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
6638, 65eqbrtrd 5085 . . . 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 3411 . . . . . 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 6669 . . . . . . 7 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐵𝑥) = (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
6946, 68breq12d 5076 . . . . . 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 630 . . . . 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 3627 . . . 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 834 . . 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 765 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
74 sltval 33057 . . . 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 584 . . 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 258 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵)
77 fveq2 6669 . . . . 5 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ( bday 𝑥) = ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
7877eleq1d 2902 . . . 4 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (( bday 𝑥) ∈ ( bday 𝐴) ↔ ( bday ‘(𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})) ∈ ( bday 𝐴)))
79 breq2 5067 . . . 4 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴 <s 𝑥𝐴 <s (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
80 breq1 5066 . . . 4 (𝑥 = (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝑥 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) <s 𝐵))
8178, 79, 803anbi123d 1429 . . 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 3627 . 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 1366 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 207   ∧ wa 396   ∨ w3o 1080   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  {crab 3147   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295  {ctp 4568  ⟨cop 4570  ∩ cint 4874   class class class wbr 5063  dom cdm 5554   ↾ cres 5556  Oncon0 6190  ⟶wf 6350  –onto→wfo 6352  ‘cfv 6354  1oc1o 8091  2oc2o 8092   No csur 33050
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