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Theorem infinfnum 35757

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 10392	. . 3 ⊢ (𝐴 ∈ dom card → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
2	1	con2bid 357	. 2 ⊢ (𝐴 ∈ dom card → (ω ≼ 𝐴 ↔ ¬ 𝐴 ∈ Fin^IV))
3		fin45 10470	. . . . . 6 ⊢ (𝐴 ∈ Fin^IV → 𝐴 ∈ Fin^V)
4		fin56 10471	. . . . . 6 ⊢ (𝐴 ∈ Fin^V → 𝐴 ∈ Fin^VI)
5		fin67 10473	. . . . . 6 ⊢ (𝐴 ∈ Fin^VI → 𝐴 ∈ Fin^VII)
6	3, 4, 5	3syl 19	. . . . 5 ⊢ (𝐴 ∈ Fin^IV → 𝐴 ∈ Fin^VII)
7		fin71num 10475	. . . . 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 10491	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^II)
10		fin23 10467	. . . . 5 ⊢ (𝐴 ∈ Fin^II → 𝐴 ∈ Fin^III)
11		fin34 10468	. . . . 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 2145 class class class wbr 5103 dom cdm 5651 ωcom 7877 ≼ cdom 8971 Fincfn 8973 cardccrd 10016 Fin^IIcfin2 10357 Fin^IVcfin4 10358 Fin^IIIcfin3 10359 Fin^Vcfin5 10360 Fin^VIcfin6 10361 Fin^VIIcfin7 10362

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-rpss 7739 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-seqom 8458 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-oi 9504 df-wdom 9559 df-dju 9982 df-card 10020 df-fin2 10364 df-fin4 10365 df-fin3 10366 df-fin5 10367 df-fin6 10368 df-fin7 10369

This theorem is used by: acwer1prc 35760

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