Theorem isnumi 6362

Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
isnumi	⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)

Proof of Theorem isnumi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 3767	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
2	1	rspcev 2656	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		intexrabim 3907	. . . 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 6227	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	simprd 107	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
7		breq2 3768	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐵))
8	7	rabbidv 2549	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
9	8	inteqd 3620	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
10	9	eleq1d 2106	. . . . . 6 ⊢ (𝑥 = 𝐵 → (∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V))
11	10	elrab3 2699	. . . . 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 262	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → (𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V))
14	4, 13	mpbird 156	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V})
15		df-card 6360	. . 3 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
16	15	dmmpt 4816	. 2 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
17	14, 16	syl6eleqr 2131	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  {crab 2310  Vcvv 2557  ∩ cint 3615   class class class wbr 3764  Oncon0 4100  dom cdm 4345   ≈ 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-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

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-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-en 6222  df-card 6360

This theorem is referenced by:  finnum  6363  onenon  6364