infpssALT - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > infpssALT		Structured version Visualization version GIF version

Theorem infpssALT 9738

Description: Alternate proof of infpss 9642, shorter but requiring Replacement (ax-rep 5193). (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 9737	. 2 ⊢ ¬ ω ∈ Fin^IV
2		reldom 8518	. . . . 5 ⊢ Rel ≼
3	2	brrelex2i 5612	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
4		isfin4 9722	. . . 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 9736	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ ω ≼ 𝐴) → ω ∈ Fin^IV)
7	6	expcom 416	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ Fin^IV → ω ∈ Fin^IV))
8	5, 7	sylbird 262	. 2 ⊢ (ω ≼ 𝐴 → (¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ∈ Fin^IV))
9	1, 8	mt3i 151	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1779 ∈ wcel 2113 Vcvv 3497 ⊊ wpss 3940 class class class wbr 5069 ωcom 7583 ≈ cen 8509 ≼ cdom 8510 Fin^IVcfin4 9705

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-er 8292 df-en 8513 df-dom 8514 df-fin4 9712

This theorem is referenced by: (None)

W3C validator