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 10347

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 10330	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^IV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)))
2		infpssr 10341	. . . . 5 ⊢ ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
3	2	exlimiv 1925	. . . 4 ⊢ (∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) → ω ≼ 𝐴)
4		infpss 10250	. . . 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 394 ∃wex 1773 ∈ wcel 2098 ⊊ wpss 3950 class class class wbr 5152 ωcom 7878 ≈ cen 8969 ≼ cdom 8970 Fin^IVcfin4 10313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-fin4 10320

This theorem is referenced by: isfin4p1 10348 fin23lem41 10385 isfin32i 10398 isfin1-2 10418 fin34 10423 fin41 10477 gchinf 10690

W3C validator