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Mirrors > Home > ILE Home > Th. List > cardcl | GIF version |
Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
cardcl | ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-card 6711 | . . . 4 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}) | |
2 | 1 | a1i 9 | . . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})) |
3 | breq2 3815 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴)) | |
4 | 3 | rabbidv 2601 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
5 | 4 | inteqd 3667 | . . . 4 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
6 | 5 | adantl 271 | . . 3 ⊢ ((∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
7 | encv 6393 | . . . . 5 ⊢ (𝑦 ≈ 𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V)) | |
8 | 7 | simprd 112 | . . . 4 ⊢ (𝑦 ≈ 𝐴 → 𝐴 ∈ V) |
9 | 8 | rexlimivw 2479 | . . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝐴 ∈ V) |
10 | intexrabim 3954 | . . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) | |
11 | 2, 6, 9, 10 | fvmptd 5330 | . 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
12 | onintrab2im 4298 | . 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ On) | |
13 | 11, 12 | eqeltrd 2159 | 1 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ∃wrex 2354 {crab 2357 Vcvv 2612 ∩ cint 3662 class class class wbr 3811 ↦ cmpt 3865 Oncon0 4154 ‘cfv 4969 ≈ cen 6385 cardccrd 6710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-suc 4162 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-en 6388 df-card 6711 |
This theorem is referenced by: (None) |
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