isfin4-2 - Metamath Proof Explorer

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Theorem isfin4-2 10001

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 9984	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
2		infpssr 9995	. . . . 5 ⊢ ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
3	2	exlimiv 1934	. . . 4 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
4		infpss 9904	. . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))
5	3, 4	impbii 208	. . 3 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ω ≼ 𝐴)
6	5	notbii 319	. 2 ⊢ (¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ¬ ω ≼ 𝐴)
7	1, 6	bitrdi 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1783 ∈ wcel 2108 ⊊ wpss 3884 class class class wbr 5070 ωcom 7687 ≈ cen 8688 ≼ cdom 8689 Fin^IVcfin4 9967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fin4 9974

This theorem is referenced by: isfin4p1 10002 fin23lem41 10039 isfin32i 10052 isfin1-2 10072 fin34 10077 fin41 10131 gchinf 10344

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