![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cardval3ex | GIF version |
Description: The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
cardval3ex | ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 6543 | . . . 4 ⊢ (𝑥 ≈ 𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V)) | |
2 | 1 | simprd 113 | . . 3 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V) |
3 | 2 | rexlimivw 2498 | . 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V) |
4 | breq1 3870 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴)) | |
5 | 4 | cbvrexv 2605 | . . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) |
6 | intexrabim 4010 | . . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) | |
7 | 5, 6 | sylbir 134 | . 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) |
8 | breq2 3871 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑦 ≈ 𝑧 ↔ 𝑦 ≈ 𝐴)) | |
9 | 8 | rabbidv 2622 | . . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
10 | 9 | inteqd 3715 | . . 3 ⊢ (𝑧 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
11 | df-card 6905 | . . 3 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) | |
12 | 10, 11 | fvmptg 5415 | . 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
13 | 3, 7, 12 | syl2anc 404 | 1 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 ∃wrex 2371 {crab 2374 Vcvv 2633 ∩ cint 3710 class class class wbr 3867 Oncon0 4214 ‘cfv 5049 ≈ cen 6535 cardccrd 6904 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-en 6538 df-card 6905 |
This theorem is referenced by: oncardval 6911 carden2bex 6914 |
Copyright terms: Public domain | W3C validator |