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Theorem nobndlem4 31094
 Description: Lemma for nobndup 31099 and nobnddown 31100. 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 31079 . . 3 ( bday 𝐴) ∈ On
2 nobndlem4.1 . . . . 5 𝑋 ∈ {1𝑜, 2𝑜}
32nosgnn0i 31056 . . . 4 ∅ ≠ 𝑋
4 fvnobday 31081 . . . . 5 (𝐴 No → (𝐴‘( bday 𝐴)) = ∅)
54neeq1d 2841 . . . 4 (𝐴 No → ((𝐴‘( bday 𝐴)) ≠ 𝑋 ↔ ∅ ≠ 𝑋))
63, 5mpbiri 247 . . 3 (𝐴 No → (𝐴‘( bday 𝐴)) ≠ 𝑋)
7 fveq2 6103 . . . . 5 (𝑥 = ( bday 𝐴) → (𝐴𝑥) = (𝐴‘( bday 𝐴)))
87neeq1d 2841 . . . 4 (𝑥 = ( bday 𝐴) → ((𝐴𝑥) ≠ 𝑋 ↔ (𝐴‘( bday 𝐴)) ≠ 𝑋))
98rspcev 3282 . . 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 6894 . 2 (∃𝑥 ∈ On (𝐴𝑥) ≠ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)
1210, 11sylib 207 1 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  ∅c0 3874  {cpr 4127  ∩ cint 4410  Oncon0 5640  ‘cfv 5804  1𝑜c1o 7440  2𝑜c2o 7441   No csur 31037   bday cbday 31039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-2o 7448  df-no 31040  df-bday 31042 This theorem is referenced by:  nobndup  31099  nobnddown  31100
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