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Mirrors > Home > MPE Home > Th. List > winalim | Structured version Visualization version GIF version |
Description: A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winalim | ⊢ (𝐴 ∈ Inaccw → Lim 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | winainf 10443 | . 2 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) | |
2 | winacard 10441 | . . 3 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) | |
3 | cardlim 9723 | . . . 4 ⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) | |
4 | sseq2 3952 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ 𝐴)) | |
5 | limeq 6276 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (Lim (card‘𝐴) ↔ Lim 𝐴)) | |
6 | 4, 5 | bibi12d 346 | . . . 4 ⊢ ((card‘𝐴) = 𝐴 → ((ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) ↔ (ω ⊆ 𝐴 ↔ Lim 𝐴))) |
7 | 3, 6 | mpbii 232 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 ↔ Lim 𝐴)) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ Inaccw → (ω ⊆ 𝐴 ↔ Lim 𝐴)) |
9 | 1, 8 | mpbid 231 | 1 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 Lim wlim 6265 ‘cfv 6431 ωcom 7701 cardccrd 9686 Inaccwcwina 10431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-om 7702 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-card 9690 df-cf 9692 df-wina 10433 |
This theorem is referenced by: inar1 10524 inatsk 10527 tskuni 10532 grur1a 10568 |
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