isfin4-2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isfin4-2		Structured version Visualization version GIF version

Theorem isfin4-2 10274

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 10257	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
2		infpssr 10268	. . . . 5 ⊢ ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
3	2	exlimiv 1930	. . . 4 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
4		infpss 10176	. . . 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 2109 ⊊ wpss 3918 class class class wbr 5110 ωcom 7845 ≈ cen 8918 ≼ cdom 8919 Fin^IVcfin4 10240

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-fin4 10247

This theorem is referenced by: isfin4p1 10275 fin23lem41 10312 isfin32i 10325 isfin1-2 10345 fin34 10350 fin41 10404 gchinf 10617

W3C validator