infpssALT - Metamath Proof Explorer

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Theorem infpssALT 10311

Description: Alternate proof of infpss 10215, shorter but requiring Replacement (ax-rep 5286). (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(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))

Distinct variable group: 𝑥,𝐴

**Proof of Theorem infpssALT**
Step	Hyp	Ref	Expression
1		ominf4 10310	. 2 ⊢ ¬ ω ∈ Fin^IV
2		reldom 8948	. . . . 5 ⊢ Rel ≼
3	2	brrelex2i 5734	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		isfin4 10295	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
5	3, 4	syl 17	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
6		domfin4 10309	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ ω ≼ 𝐴) → ω ∈ Fin^IV)
7	6	expcom 413	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ Fin^IV → ω ∈ Fin^IV))
8	5, 7	sylbird 259	. 2 ⊢ (ω ≼ 𝐴 → (¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ∈ Fin^IV))
9	1, 8	mt3i 149	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 ⊊ wpss 3950 class class class wbr 5149 ωcom 7858 ≈ cen 8939 ≼ cdom 8940 Fin^IVcfin4 10278

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7859 df-er 8706 df-en 8943 df-dom 8944 df-fin4 10285

This theorem is referenced by: (None)

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