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Theorem nobndlem4 31555
Description: Lemma for nobndup 31560 and nobnddown 31561. The infimum of the class of all ordinals such that 𝐴 is not 𝑋 is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
Hypothesis
Ref Expression
nobndlem4.1 𝑋 ∈ {1𝑜, 2𝑜}
Assertion
Ref Expression
nobndlem4 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem nobndlem4
StepHypRef Expression
1 bdayelon 31540 . . 3 ( bday 𝐴) ∈ On
2 nobndlem4.1 . . . . 5 𝑋 ∈ {1𝑜, 2𝑜}
32nosgnn0i 31510 . . . 4 ∅ ≠ 𝑋
4 fvnobday 31542 . . . . 5 (𝐴 No → (𝐴‘( bday 𝐴)) = ∅)
54neeq1d 2849 . . . 4 (𝐴 No → ((𝐴‘( bday 𝐴)) ≠ 𝑋 ↔ ∅ ≠ 𝑋))
63, 5mpbiri 248 . . 3 (𝐴 No → (𝐴‘( bday 𝐴)) ≠ 𝑋)
7 fveq2 6148 . . . . 5 (𝑥 = ( bday 𝐴) → (𝐴𝑥) = (𝐴‘( bday 𝐴)))
87neeq1d 2849 . . . 4 (𝑥 = ( bday 𝐴) → ((𝐴𝑥) ≠ 𝑋 ↔ (𝐴‘( bday 𝐴)) ≠ 𝑋))
98rspcev 3295 . . 3 ((( bday 𝐴) ∈ On ∧ (𝐴‘( bday 𝐴)) ≠ 𝑋) → ∃𝑥 ∈ On (𝐴𝑥) ≠ 𝑋)
101, 6, 9sylancr 694 . 2 (𝐴 No → ∃𝑥 ∈ On (𝐴𝑥) ≠ 𝑋)
11 onintrab2 6949 . 2 (∃𝑥 ∈ On (𝐴𝑥) ≠ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)
1210, 11sylib 208 1 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  c0 3891  {cpr 4150   cint 4440  Oncon0 5682  cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499   No csur 31491   bday cbday 31493
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1o 7505  df-2o 7506  df-no 31494  df-bday 31496
This theorem is referenced by:  nobndup  31560  nobnddown  31561
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