Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cardval3 | Structured version Visualization version GIF version |
Description: An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
cardval3 | ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3416 | . 2 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V) | |
2 | isnum2 9448 | . . . 4 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) | |
3 | rabn0 4275 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) | |
4 | intex 5206 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) | |
5 | 2, 3, 4 | 3bitr2i 302 | . . 3 ⊢ (𝐴 ∈ dom card ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) |
6 | 5 | biimpi 219 | . 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) |
7 | breq2 5035 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴)) | |
8 | 7 | rabbidv 3381 | . . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
9 | 8 | inteqd 4842 | . . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
10 | df-card 9442 | . . 3 ⊢ card = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦}) | |
11 | 9, 10 | fvmptg 6774 | . 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
12 | 1, 6, 11 | syl2anc 587 | 1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∃wrex 3054 {crab 3057 Vcvv 3398 ∅c0 4212 ∩ cint 4837 class class class wbr 5031 dom cdm 5526 Oncon0 6173 ‘cfv 6340 ≈ cen 8553 cardccrd 9438 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7480 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3683 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-ord 6176 df-on 6177 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-en 8557 df-card 9442 |
This theorem is referenced by: cardid2 9456 oncardval 9458 cardidm 9462 cardne 9468 cardval 10047 |
Copyright terms: Public domain | W3C validator |