isfin4-2 - Metamath Proof Explorer

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Theorem isfin4-2 10354

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 10337	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
2		infpssr 10348	. . . . 5 ⊢ ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
3	2	exlimiv 1930	. . . 4 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
4		infpss 10256	. . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))
5	3, 4	impbii 209	. . 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 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ⊊ wpss 3952 class class class wbr 5143 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 Fin^IVcfin4 10320

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-fin4 10327

This theorem is referenced by: isfin4p1 10355 fin23lem41 10392 isfin32i 10405 isfin1-2 10425 fin34 10430 fin41 10484 gchinf 10697

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