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Theorem infinfnum 35825

Description: Equivalence between two infiniteness criteria for numerable sets. (Contributed by BTernaryTau, 15-Jul-2026.)

**Assertion**
Ref	Expression
infinfnum	⊢ (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))

**Proof of Theorem infinfnum**
Step	Hyp	Ref	Expression
1		isfin4-2 10399	. . 3 ⊢ (𝐴 ∈ dom card → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
2	1	con2bid 357	. 2 ⊢ (𝐴 ∈ dom card → (ω ≼ 𝐴 ↔ ¬ 𝐴 ∈ Fin^IV))
3		fin45 10477	. . . . . 6 ⊢ (𝐴 ∈ Fin^IV → 𝐴 ∈ Fin^V)
4		fin56 10478	. . . . . 6 ⊢ (𝐴 ∈ Fin^V → 𝐴 ∈ Fin^VI)
5		fin67 10480	. . . . . 6 ⊢ (𝐴 ∈ Fin^VI → 𝐴 ∈ Fin^VII)
6	3, 4, 5	3syl 19	. . . . 5 ⊢ (𝐴 ∈ Fin^IV → 𝐴 ∈ Fin^VII)
7		fin71num 10482	. . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 ∈ Fin^VII ↔ 𝐴 ∈ Fin))
8	6, 7	imbitrid 247	. . . 4 ⊢ (𝐴 ∈ dom card → (𝐴 ∈ Fin^IV → 𝐴 ∈ Fin))
9		fin12 10498	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^II)
10		fin23 10474	. . . . 5 ⊢ (𝐴 ∈ Fin^II → 𝐴 ∈ Fin^III)
11		fin34 10475	. . . . 5 ⊢ (𝐴 ∈ Fin^III → 𝐴 ∈ Fin^IV)
12	9, 10, 11	3syl 19	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^IV)
13	8, 12	impbid1 228	. . 3 ⊢ (𝐴 ∈ dom card → (𝐴 ∈ Fin^IV ↔ 𝐴 ∈ Fin))
14	13	notbid 321	. 2 ⊢ (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin^IV ↔ ¬ 𝐴 ∈ Fin))
15	2, 14	bitr2d 283	1 ⊢ (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2144 class class class wbr 5102 dom cdm 5651 ωcom 7881 ≼ cdom 8979 Fincfn 8981 cardccrd 10023 Fin^IIcfin2 10364 Fin^IVcfin4 10365 Fin^IIIcfin3 10366 Fin^Vcfin5 10367 Fin^VIcfin6 10368 Fin^VIIcfin7 10369

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-rpss 7743 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-seqom 8466 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-oi 9511 df-wdom 9566 df-dju 9989 df-card 10027 df-fin2 10371 df-fin4 10372 df-fin3 10373 df-fin5 10374 df-fin6 10375 df-fin7 10376

This theorem is used by: acwer1prc 35828 j0ltaleph 35959 jltaleph 35977

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