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Theorem nodenselem4 31582
Description: Lemma for nodense 31587. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem4 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem4
StepHypRef Expression
1 ssrab2 3671 . 2 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ On
2 sltirr 31565 . . . . . . 7 (𝐴 No → ¬ 𝐴 <s 𝐴)
3 breq2 4622 . . . . . . . . 9 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
43biimprcd 240 . . . . . . . 8 (𝐴 <s 𝐵 → (𝐴 = 𝐵𝐴 <s 𝐴))
54con3d 148 . . . . . . 7 (𝐴 <s 𝐵 → (¬ 𝐴 <s 𝐴 → ¬ 𝐴 = 𝐵))
62, 5syl5com 31 . . . . . 6 (𝐴 No → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
76adantr 481 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
8 nofnbday 31541 . . . . . . . 8 (𝐴 No 𝐴 Fn ( bday 𝐴))
9 nofnbday 31541 . . . . . . . 8 (𝐵 No 𝐵 Fn ( bday 𝐵))
10 eqfnfv2 6273 . . . . . . . 8 ((𝐴 Fn ( bday 𝐴) ∧ 𝐵 Fn ( bday 𝐵)) → (𝐴 = 𝐵 ↔ (( bday 𝐴) = ( bday 𝐵) ∧ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎))))
118, 9, 10syl2an 494 . . . . . . 7 ((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (( bday 𝐴) = ( bday 𝐵) ∧ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎))))
1211notbid 308 . . . . . 6 ((𝐴 No 𝐵 No ) → (¬ 𝐴 = 𝐵 ↔ ¬ (( bday 𝐴) = ( bday 𝐵) ∧ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎))))
13 ianor 509 . . . . . . 7 (¬ (( bday 𝐴) = ( bday 𝐵) ∧ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎)) ↔ (¬ ( bday 𝐴) = ( bday 𝐵) ∨ ¬ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎)))
14 bdayelon 31578 . . . . . . . . . . . 12 ( bday 𝐴) ∈ On
1514onordi 5796 . . . . . . . . . . 11 Ord ( bday 𝐴)
16 bdayelon 31578 . . . . . . . . . . . 12 ( bday 𝐵) ∈ On
1716onordi 5796 . . . . . . . . . . 11 Ord ( bday 𝐵)
18 ordtri3 5723 . . . . . . . . . . 11 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → (( bday 𝐴) = ( bday 𝐵) ↔ ¬ (( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴))))
1915, 17, 18mp2an 707 . . . . . . . . . 10 (( bday 𝐴) = ( bday 𝐵) ↔ ¬ (( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)))
2019con2bii 347 . . . . . . . . 9 ((( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)) ↔ ¬ ( bday 𝐴) = ( bday 𝐵))
21 nodenselem3 31581 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
22 nodenselem3 31581 . . . . . . . . . . . 12 ((𝐵 No 𝐴 No ) → (( bday 𝐵) ∈ ( bday 𝐴) → ∃𝑎 ∈ On (𝐵𝑎) ≠ (𝐴𝑎)))
23 necom 2843 . . . . . . . . . . . . 13 ((𝐵𝑎) ≠ (𝐴𝑎) ↔ (𝐴𝑎) ≠ (𝐵𝑎))
2423rexbii 3035 . . . . . . . . . . . 12 (∃𝑎 ∈ On (𝐵𝑎) ≠ (𝐴𝑎) ↔ ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
2522, 24syl6ib 241 . . . . . . . . . . 11 ((𝐵 No 𝐴 No ) → (( bday 𝐵) ∈ ( bday 𝐴) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
2625ancoms 469 . . . . . . . . . 10 ((𝐴 No 𝐵 No ) → (( bday 𝐵) ∈ ( bday 𝐴) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
2721, 26jaod 395 . . . . . . . . 9 ((𝐴 No 𝐵 No ) → ((( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
2820, 27syl5bir 233 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (¬ ( bday 𝐴) = ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
29 rexnal 2990 . . . . . . . . . 10 (∃𝑎 ∈ ( bday 𝐴) ¬ (𝐴𝑎) = (𝐵𝑎) ↔ ¬ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎))
3014onssi 6991 . . . . . . . . . . . 12 ( bday 𝐴) ⊆ On
31 ssrexv 3651 . . . . . . . . . . . 12 (( bday 𝐴) ⊆ On → (∃𝑎 ∈ ( bday 𝐴) ¬ (𝐴𝑎) = (𝐵𝑎) → ∃𝑎 ∈ On ¬ (𝐴𝑎) = (𝐵𝑎)))
3230, 31ax-mp 5 . . . . . . . . . . 11 (∃𝑎 ∈ ( bday 𝐴) ¬ (𝐴𝑎) = (𝐵𝑎) → ∃𝑎 ∈ On ¬ (𝐴𝑎) = (𝐵𝑎))
33 df-ne 2791 . . . . . . . . . . . 12 ((𝐴𝑎) ≠ (𝐵𝑎) ↔ ¬ (𝐴𝑎) = (𝐵𝑎))
3433rexbii 3035 . . . . . . . . . . 11 (∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎) ↔ ∃𝑎 ∈ On ¬ (𝐴𝑎) = (𝐵𝑎))
3532, 34sylibr 224 . . . . . . . . . 10 (∃𝑎 ∈ ( bday 𝐴) ¬ (𝐴𝑎) = (𝐵𝑎) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
3629, 35sylbir 225 . . . . . . . . 9 (¬ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
3736a1i 11 . . . . . . . 8 ((𝐴 No 𝐵 No ) → (¬ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
3828, 37jaod 395 . . . . . . 7 ((𝐴 No 𝐵 No ) → ((¬ ( bday 𝐴) = ( bday 𝐵) ∨ ¬ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎)) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
3913, 38syl5bi 232 . . . . . 6 ((𝐴 No 𝐵 No ) → (¬ (( bday 𝐴) = ( bday 𝐵) ∧ ∀𝑎 ∈ ( bday 𝐴)(𝐴𝑎) = (𝐵𝑎)) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
4012, 39sylbid 230 . . . . 5 ((𝐴 No 𝐵 No ) → (¬ 𝐴 = 𝐵 → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
417, 40syld 47 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
4241imp 445 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
43 rabn0 3937 . . 3 ({𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅ ↔ ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
4442, 43sylibr 224 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅)
45 oninton 6954 . 2 (({𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ On ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ≠ ∅) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
461, 44, 45sylancr 694 1 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3559  c0 3896   cint 4445   class class class wbr 4618  Ord word 5686  Oncon0 5687   Fn wfn 5847  cfv 5852   No csur 31529   <s cslt 31530   bday cbday 31531
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-1o 7512  df-2o 7513  df-no 31532  df-slt 31533  df-bday 31534
This theorem is referenced by:  nodenselem5  31583  nodenselem6  31584  nodenselem7  31585  nodense  31587
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