isfin4-2 - Metamath Proof Explorer

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Theorem isfin4-2 10175

Description: Alternate definition of IV-finite sets: they lack a denumerable subset. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
isfin4-2	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))

Proof of Theorem isfin4-2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfin4 10158	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
2		infpssr 10169	. . . . 5 ⊢ ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
3	2	exlimiv 1933	. . . 4 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
4		infpss 10078	. . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))
5	3, 4	impbii 208	. . 3 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ω ≼ 𝐴)
6	5	notbii 320	. 2 ⊢ (¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ¬ ω ≼ 𝐴)
7	1, 6	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∃wex 1781 ∈ wcel 2106 ⊊ wpss 3902 class class class wbr 5096 ωcom 7784 ≈ cen 8805 ≼ cdom 8806 Fin^IVcfin4 10141

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4857 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-ov 7344 df-om 7785 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-er 8573 df-en 8809 df-dom 8810 df-fin4 10148

This theorem is referenced by: isfin4p1 10176 fin23lem41 10213 isfin32i 10226 isfin1-2 10246 fin34 10251 fin41 10305 gchinf 10518

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