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Theorem infpssALT 9080
 Description: Alternate proof of infpss 8984, shorter but requiring Replacement (ax-rep 4736). (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 9079 . 2 ¬ ω ∈ FinIV
2 reldom 7906 . . . . 5 Rel ≼
32brrelex2i 5124 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
4 isfin4 9064 . . . 4 (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
53, 4syl 17 . . 3 (ω ≼ 𝐴 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐴𝑥𝐴)))
6 domfin4 9078 . . . 4 ((𝐴 ∈ FinIV ∧ ω ≼ 𝐴) → ω ∈ FinIV)
76expcom 451 . . 3 (ω ≼ 𝐴 → (𝐴 ∈ FinIV → ω ∈ FinIV))
85, 7sylbird 250 . 2 (ω ≼ 𝐴 → (¬ ∃𝑥(𝑥𝐴𝑥𝐴) → ω ∈ FinIV))
91, 8mt3i 141 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∃wex 1701   ∈ wcel 1992  Vcvv 3191   ⊊ wpss 3561   class class class wbr 4618  ωcom 7013   ≈ cen 7897   ≼ cdom 7898  FinIVcfin4 9047 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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903 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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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-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 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-er 7688  df-en 7901  df-dom 7902  df-fin4 9054 This theorem is referenced by: (None)
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