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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 6721 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴) | |
2 | breq1 3979 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴)) | |
3 | 2 | rspcev 2825 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
4 | 1, 3 | mpdan 418 | . 2 ⊢ (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
5 | cardval3ex 7132 | . 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 1342 ∈ wcel 2135 ∃wrex 2443 {crab 2446 ∩ cint 3818 class class class wbr 3976 Oncon0 4335 ‘cfv 5182 ≈ cen 6695 cardccrd 7126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-en 6698 df-card 7127 |
This theorem is referenced by: cardonle 7134 |
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