Theorem isnumi 7260

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 4037	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
2	1	rspcev 2868	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		intexrabim 4187	. . . 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 6814	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	simprd 114	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
7		breq2 4038	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐵))
8	7	rabbidv 2752	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
9	8	inteqd 3880	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
10	9	eleq1d 2265	. . . . . 6 ⊢ (𝑥 = 𝐵 → (∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V))
11	10	elrab3 2921	. . . . 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 277	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → (𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V))
14	4, 13	mpbird 167	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V})
15		df-card 7257	. . 3 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
16	15	dmmpt 5166	. 2 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
17	14, 16	eleqtrrdi 2290	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  {crab 2479  Vcvv 2763  ∩ cint 3875   class class class wbr 4034  Oncon0 4399  dom cdm 4664   ≈ cen 6806  cardccrd 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-xp 4670  df-rel 4671  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-en 6809  df-card 7257

This theorem is referenced by:  finnum  7261  onenon  7262