Theorem isnumi 7478

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 4112	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
2	1	rspcev 2921	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		intexrabim 4265	. . . 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 6981	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	simprd 114	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
7		breq2 4113	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐵))
8	7	rabbidv 2802	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
9	8	inteqd 3954	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
10	9	eleq1d 2301	. . . . . 6 ⊢ (𝑥 = 𝐵 → (∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V))
11	10	elrab3 2974	. . . . 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 7475	. . 3 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
16	15	dmmpt 5258	. 2 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
17	14, 16	eleqtrrdi 2326	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  {crab 2524  Vcvv 2813  ∩ cint 3949   class class class wbr 4109  Oncon0 4484  dom cdm 4749   ≈ cen 6973  cardccrd 7473

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-en 6976  df-card 7475

This theorem is referenced by:  finnum  7479  onenon  7480