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Mirrors > Home > ILE Home > Th. List > isnumi | GIF version |
Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
isnumi | ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3870 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵)) | |
2 | 1 | rspcev 2736 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑦 ∈ On 𝑦 ≈ 𝐵) |
3 | intexrabim 4010 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V) |
5 | encv 6543 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 5 | simprd 113 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
7 | breq2 3871 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐵)) | |
8 | 7 | rabbidv 2622 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵}) |
9 | 8 | inteqd 3715 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵}) |
10 | 9 | eleq1d 2163 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V)) |
11 | 10 | elrab3 2786 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V)) |
12 | 6, 11 | syl 14 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V)) |
13 | 12 | adantl 272 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → (𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V)) |
14 | 4, 13 | mpbird 166 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}) |
15 | df-card 6905 | . . 3 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}) | |
16 | 15 | dmmpt 4960 | . 2 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} |
17 | 14, 16 | syl6eleqr 2188 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∈ wcel 1445 ∃wrex 2371 {crab 2374 Vcvv 2633 ∩ cint 3710 class class class wbr 3867 Oncon0 4214 dom cdm 4467 ≈ 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-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-xp 4473 df-rel 4474 df-cnv 4475 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-en 6538 df-card 6905 |
This theorem is referenced by: finnum 6908 onenon 6909 |
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