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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  ontowfo 6352  cfv 6354  1oc1o 8091  2oc2o 8092   No csur 33050   <s cslt 33051   bday cbday 33052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-1o 8098  df-2o 8099  df-no 33053  df-slt 33054  df-bday 33055
This theorem is referenced by:  nocvxminlem  33150
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