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Theorem infpssALT 9733
 Description: Alternate proof of infpss 9637, shorter but requiring Replacement (ax-rep 5176). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
infpssALT (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem infpssALT
StepHypRef Expression
1 ominf4 9732 . 2 ¬ ω ∈ FinIV
2 reldom 8511 . . . . 5 Rel ≼
32brrelex2i 5596 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
4 isfin4 9717 . . . 4 (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
53, 4syl 17 . . 3 (ω ≼ 𝐴 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
6 domfin4 9731 . . . 4 ((𝐴 ∈ FinIV ∧ ω ≼ 𝐴) → ω ∈ FinIV)
76expcom 417 . . 3 (ω ≼ 𝐴 → (𝐴 ∈ FinIV → ω ∈ FinIV))
85, 7sylbird 263 . 2 (ω ≼ 𝐴 → (¬ ∃𝑥(𝑥𝐴𝑥𝐴) → ω ∈ FinIV))
91, 8mt3i 151 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2115  Vcvv 3480   ⊊ wpss 3920   class class class wbr 5052  ωcom 7574   ≈ cen 8502   ≼ cdom 8503  FinIVcfin4 9700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-er 8285  df-en 8506  df-dom 8507  df-fin4 9707 This theorem is referenced by: (None)
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