Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > oncardval | GIF version |
Description: The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
oncardval | ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrefg 6244 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴) | |
2 | breq1 3767 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴)) | |
3 | 2 | rspcev 2656 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
4 | 1, 3 | mpdan 398 | . 2 ⊢ (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
5 | cardval3ex 6365 | . 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) | |
6 | 4, 5 | syl 14 | 1 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 {crab 2310 ∩ cint 3615 class class class wbr 3764 Oncon0 4100 ‘cfv 4902 ≈ cen 6219 cardccrd 6359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-en 6222 df-card 6360 |
This theorem is referenced by: cardonle 6367 |
Copyright terms: Public domain | W3C validator |