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Theorem nodenselem5 31601
Description: Lemma for nodense 31605. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 31600 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem5
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sltirr 31583 . . . . . . . . 9 (𝐴 No → ¬ 𝐴 <s 𝐴)
2 breq2 4627 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
32notbid 308 . . . . . . . . 9 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
41, 3syl5ibcom 235 . . . . . . . 8 (𝐴 No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
54con2d 129 . . . . . . 7 (𝐴 No → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
65imp 445 . . . . . 6 ((𝐴 No 𝐴 <s 𝐵) → ¬ 𝐴 = 𝐵)
76ad2ant2rl 784 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → ¬ 𝐴 = 𝐵)
8 nofun 31556 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
9 nofun 31556 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
10 eqfunfv 6282 . . . . . . . . 9 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
118, 9, 10syl2an 494 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
1211notbid 308 . . . . . . 7 ((𝐴 No 𝐵 No ) → (¬ 𝐴 = 𝐵 ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
1312adantr 481 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → (¬ 𝐴 = 𝐵 ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
14 imnan 438 . . . . . . . . . . . 12 ((dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
1514biimpri 218 . . . . . . . . . . 11 (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) → (dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
1615impcom 446 . . . . . . . . . 10 ((dom 𝐴 = dom 𝐵 ∧ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))) → ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
17 df-ne 2791 . . . . . . . . . . . . 13 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
1817rexbii 3036 . . . . . . . . . . . 12 (∃𝑥 ∈ dom 𝐴(𝐴𝑥) ≠ (𝐵𝑥) ↔ ∃𝑥 ∈ dom 𝐴 ¬ (𝐴𝑥) = (𝐵𝑥))
19 rexnal 2991 . . . . . . . . . . . 12 (∃𝑥 ∈ dom 𝐴 ¬ (𝐴𝑥) = (𝐵𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
2018, 19bitri 264 . . . . . . . . . . 11 (∃𝑥 ∈ dom 𝐴(𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
21 nodenselem4 31600 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
22 eloni 5702 . . . . . . . . . . . . . . 15 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
2321, 22syl 17 . . . . . . . . . . . . . 14 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
2423adantr 481 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
25 nodmord 31560 . . . . . . . . . . . . . 14 (𝐴 No → Ord dom 𝐴)
2625ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → Ord dom 𝐴)
27 nodmon 31557 . . . . . . . . . . . . . . . . . . 19 (𝐴 No → dom 𝐴 ∈ On)
28 onelon 5717 . . . . . . . . . . . . . . . . . . 19 ((dom 𝐴 ∈ On ∧ 𝑥 ∈ dom 𝐴) → 𝑥 ∈ On)
2927, 28sylan 488 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑥 ∈ dom 𝐴) → 𝑥 ∈ On)
3029ex 450 . . . . . . . . . . . . . . . . 17 (𝐴 No → (𝑥 ∈ dom 𝐴𝑥 ∈ On))
3130ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (𝑥 ∈ dom 𝐴𝑥 ∈ On))
3231anim1d 587 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → ((𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥)) → (𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥))))
3332imp 445 . . . . . . . . . . . . . 14 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → (𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥)))
34 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝐴𝑎) = (𝐴𝑥))
35 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝐵𝑎) = (𝐵𝑥))
3634, 35neeq12d 2851 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴𝑥) ≠ (𝐵𝑥)))
3736intminss 4475 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥)
3833, 37syl 17 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥)
39 simprl 793 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → 𝑥 ∈ dom 𝐴)
40 ordtr2 5737 . . . . . . . . . . . . . 14 ((Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∧ Ord dom 𝐴) → (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥𝑥 ∈ dom 𝐴) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4140imp 445 . . . . . . . . . . . . 13 (((Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∧ Ord dom 𝐴) ∧ ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥𝑥 ∈ dom 𝐴)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
4224, 26, 38, 39, 41syl22anc 1324 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
4342rexlimdvaa 3027 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (∃𝑥 ∈ dom 𝐴(𝐴𝑥) ≠ (𝐵𝑥) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4420, 43syl5bir 233 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4516, 44syl5 34 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → ((dom 𝐴 = dom 𝐵 ∧ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4645exp4b 631 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → (dom 𝐴 = dom 𝐵 → (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))))
4746com23 86 . . . . . . 7 ((𝐴 No 𝐵 No ) → (dom 𝐴 = dom 𝐵 → (𝐴 <s 𝐵 → (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))))
4847imp32 449 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4913, 48sylbid 230 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → (¬ 𝐴 = 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
507, 49mpd 15 . . . 4 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
5150ex 450 . . 3 ((𝐴 No 𝐵 No ) → ((dom 𝐴 = dom 𝐵𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
52 bdayval 31555 . . . . 5 (𝐴 No → ( bday 𝐴) = dom 𝐴)
53 bdayval 31555 . . . . 5 (𝐵 No → ( bday 𝐵) = dom 𝐵)
5452, 53eqeqan12d 2637 . . . 4 ((𝐴 No 𝐵 No ) → (( bday 𝐴) = ( bday 𝐵) ↔ dom 𝐴 = dom 𝐵))
5554anbi1d 740 . . 3 ((𝐴 No 𝐵 No ) → ((( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵) ↔ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)))
5652eleq2d 2684 . . . 4 (𝐴 No → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) ↔ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
5756adantr 481 . . 3 ((𝐴 No 𝐵 No ) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) ↔ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
5851, 55, 573imtr4d 283 . 2 ((𝐴 No 𝐵 No ) → ((( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)))
5958imp 445 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  {crab 2912  wss 3560   cint 4447   class class class wbr 4623  dom cdm 5084  Ord word 5691  Oncon0 5692  Fun wfun 5851  cfv 5857   No csur 31547   <s cslt 31548   bday cbday 31549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-1o 7520  df-2o 7521  df-no 31550  df-slt 31551  df-bday 31552
This theorem is referenced by:  nodenselem6  31602  nodenselem8  31604  nodense  31605
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