![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10733 | . 2 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) | |
2 | winacard 10731 | . . 3 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) | |
3 | cardlim 10011 | . . . 4 ⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) | |
4 | sseq2 4005 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ 𝐴)) | |
5 | limeq 6387 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (Lim (card‘𝐴) ↔ Lim 𝐴)) | |
6 | 4, 5 | bibi12d 344 | . . . 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 1533 ∈ wcel 2098 ⊆ wss 3946 Lim wlim 6376 ‘cfv 6553 ωcom 7875 cardccrd 9974 Inaccwcwina 10721 |
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 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 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-om 7876 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-card 9978 df-cf 9980 df-wina 10723 |
This theorem is referenced by: inar1 10814 inatsk 10817 tskuni 10822 grur1a 10858 |
Copyright terms: Public domain | W3C validator |