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Theorem infpssALT 10382

Description: Alternate proof of infpss 10285, shorter but requiring Replacement (ax-rep 5303). (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 10381	. 2 ⊢ ¬ ω ∈ Fin^IV
2		reldom 9009	. . . . 5 ⊢ Rel ≼
3	2	brrelex2i 5757	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		isfin4 10366	. . . 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 10380	. . . 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 260	. 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 206 ∧ wa 395 ∃wex 1777 ∈ wcel 2108 Vcvv 3488 ⊊ wpss 3977 class class class wbr 5166 ωcom 7903 ≈ cen 9000 ≼ cdom 9001 Fin^IVcfin4 10349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-om 7904 df-er 8763 df-en 9004 df-dom 9005 df-fin4 10356

This theorem is referenced by: (None)

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