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Theorem nodenselem5 30887
Description: Lemma for nodense 30891. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 30886 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 30872 . . . . . . . . 9 (𝐴 No → ¬ 𝐴 <s 𝐴)
2 breq2 4578 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
32notbid 306 . . . . . . . . 9 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
41, 3syl5ibcom 233 . . . . . . . 8 (𝐴 No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
54con2d 127 . . . . . . 7 (𝐴 No → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
65imp 443 . . . . . 6 ((𝐴 No 𝐴 <s 𝐵) → ¬ 𝐴 = 𝐵)
76ad2ant2rl 780 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → ¬ 𝐴 = 𝐵)
8 nofun 30849 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
9 nofun 30849 . . . . . . . . 9 (𝐵 No → Fun 𝐵)
10 eqfunfv 6206 . . . . . . . . 9 ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
118, 9, 10syl2an 492 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
1211notbid 306 . . . . . . 7 ((𝐴 No 𝐵 No ) → (¬ 𝐴 = 𝐵 ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
1312adantr 479 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)) → (¬ 𝐴 = 𝐵 ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))))
14 imnan 436 . . . . . . . . . . . 12 ((dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
1514biimpri 216 . . . . . . . . . . 11 (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)) → (dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥)))
1615impcom 444 . . . . . . . . . 10 ((dom 𝐴 = dom 𝐵 ∧ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))) → ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
17 df-ne 2778 . . . . . . . . . . . . 13 ((𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ (𝐴𝑥) = (𝐵𝑥))
1817rexbii 3019 . . . . . . . . . . . 12 (∃𝑥 ∈ dom 𝐴(𝐴𝑥) ≠ (𝐵𝑥) ↔ ∃𝑥 ∈ dom 𝐴 ¬ (𝐴𝑥) = (𝐵𝑥))
19 rexnal 2974 . . . . . . . . . . . 12 (∃𝑥 ∈ dom 𝐴 ¬ (𝐴𝑥) = (𝐵𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
2018, 19bitri 262 . . . . . . . . . . 11 (∃𝑥 ∈ dom 𝐴(𝐴𝑥) ≠ (𝐵𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥))
21 nodenselem4 30886 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
22 eloni 5633 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
25 nodmord 30853 . . . . . . . . . . . . . 14 (𝐴 No → Ord dom 𝐴)
2625ad3antrrr 761 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → Ord dom 𝐴)
27 nodmon 30850 . . . . . . . . . . . . . . . . . . 19 (𝐴 No → dom 𝐴 ∈ On)
28 onelon 5648 . . . . . . . . . . . . . . . . . . 19 ((dom 𝐴 ∈ On ∧ 𝑥 ∈ dom 𝐴) → 𝑥 ∈ On)
2927, 28sylan 486 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑥 ∈ dom 𝐴) → 𝑥 ∈ On)
3029ex 448 . . . . . . . . . . . . . . . . 17 (𝐴 No → (𝑥 ∈ dom 𝐴𝑥 ∈ On))
3130ad2antrr 757 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (𝑥 ∈ dom 𝐴𝑥 ∈ On))
3231anim1d 585 . . . . . . . . . . . . . . 15 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → ((𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥)) → (𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥))))
3332imp 443 . . . . . . . . . . . . . 14 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → (𝑥 ∈ On ∧ (𝐴𝑥) ≠ (𝐵𝑥)))
34 fveq2 6085 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝐴𝑎) = (𝐴𝑥))
35 fveq2 6085 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝐵𝑎) = (𝐵𝑥))
3634, 35neeq12d 2839 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴𝑥) ≠ (𝐵𝑥)))
3736intminss 4429 . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → 𝑥 ∈ dom 𝐴)
40 ordtr2 5668 . . . . . . . . . . . . . 14 ((Ord {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∧ Ord dom 𝐴) → (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝑥𝑥 ∈ dom 𝐴) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4140imp 443 . . . . . . . . . . . . 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 1318 . . . . . . . . . . . 12 ((((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴𝑥) ≠ (𝐵𝑥))) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
4342rexlimdvaa 3010 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (∃𝑥 ∈ dom 𝐴(𝐴𝑥) ≠ (𝐵𝑥) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4420, 43syl5bir 231 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (¬ ∀𝑥 ∈ dom 𝐴(𝐴𝑥) = (𝐵𝑥) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
4516, 44syl5 33 . . . . . . . . 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 629 . . . . . . . 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 83 . . . . . . 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 447 . . . . . 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 228 . . . . 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 448 . . 3 ((𝐴 No 𝐵 No ) → ((dom 𝐴 = dom 𝐵𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
52 bdayval 30848 . . . . 5 (𝐴 No → ( bday 𝐴) = dom 𝐴)
53 bdayval 30848 . . . . 5 (𝐵 No → ( bday 𝐵) = dom 𝐵)
5452, 53eqeqan12d 2622 . . . 4 ((𝐴 No 𝐵 No ) → (( bday 𝐴) = ( bday 𝐵) ↔ dom 𝐴 = dom 𝐵))
5554anbi1d 736 . . 3 ((𝐴 No 𝐵 No ) → ((( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵) ↔ (dom 𝐴 = dom 𝐵𝐴 <s 𝐵)))
5652eleq2d 2669 . . . 4 (𝐴 No → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) ↔ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
5756adantr 479 . . 3 ((𝐴 No 𝐵 No ) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) ↔ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴))
5851, 55, 573imtr4d 281 . 2 ((𝐴 No 𝐵 No ) → ((( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)))
5958imp 443 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 194  wa 382   = wceq 1474  wcel 1976  wne 2776  wral 2892  wrex 2893  {crab 2896  wss 3536   cint 4401   class class class wbr 4574  dom cdm 5025  Ord word 5622  Oncon0 5623  Fun wfun 5781  cfv 5787   No csur 30840   <s cslt 30841   bday cbday 30842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-ord 5626  df-on 5627  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-1o 7421  df-2o 7422  df-no 30843  df-slt 30844  df-bday 30845
This theorem is referenced by:  nodenselem6  30888  nodenselem8  30890  nodense  30891
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