Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infinf | Structured version Visualization version GIF version |
Description: Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
infinf | ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 9094 | . . 3 ⊢ ω ∈ V | |
2 | domtri 9966 | . . 3 ⊢ ((ω ∈ V ∧ 𝐴 ∈ 𝐵) → (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)) | |
3 | 1, 2 | mpan 686 | . 2 ⊢ (𝐴 ∈ 𝐵 → (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)) |
4 | isfinite 9103 | . . 3 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | |
5 | 4 | notbii 321 | . 2 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω) |
6 | 3, 5 | syl6rbbr 291 | 1 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 ωcom 7569 ≼ cdom 8495 ≺ csdm 8496 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-ac2 9873 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-ac 9530 |
This theorem is referenced by: unirnfdomd 9977 ctbssinf 34569 pibt2 34580 |
Copyright terms: Public domain | W3C validator |