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Mirrors > Home > MPE Home > Th. List > winacard | Structured version Visualization version GIF version |
Description: A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winacard | ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elwina 9823 | . 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
2 | cardcf 9389 | . . . 4 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴) | |
3 | fveq2 6433 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴)) | |
4 | id 22 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴) | |
5 | 2, 3, 4 | 3eqtr3a 2885 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴) |
6 | 5 | 3ad2ant2 1170 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → (card‘𝐴) = 𝐴) |
7 | 1, 6 | sylbi 209 | 1 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 ∅c0 4144 class class class wbr 4873 ‘cfv 6123 ≺ csdm 8221 cardccrd 9074 cfccf 9076 Inaccwcwina 9819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-er 8009 df-en 8223 df-card 9078 df-cf 9080 df-wina 9821 |
This theorem is referenced by: winalim 9832 winalim2 9833 gchina 9836 inar1 9912 |
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