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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 9508 | . 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
2 | cardcf 9074 | . . . 4 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴) | |
3 | fveq2 6191 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴)) | |
4 | id 22 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴) | |
5 | 2, 3, 4 | 3eqtr3a 2680 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴) |
6 | 5 | 3ad2ant2 1083 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → (card‘𝐴) = 𝐴) |
7 | 1, 6 | sylbi 207 | 1 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 ≺ csdm 7954 cardccrd 8761 cfccf 8763 Inaccwcwina 9504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-card 8765 df-cf 8767 df-wina 9506 |
This theorem is referenced by: winalim 9517 winalim2 9518 gchina 9521 inar1 9597 |
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