Theorem isnumi 7385

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 4091	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
2	1	rspcev 2910	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		intexrabim 4243	. . . 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 6914	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	simprd 114	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
7		breq2 4092	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐵))
8	7	rabbidv 2791	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
9	8	inteqd 3933	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵})
10	9	eleq1d 2300	. . . . . 6 ⊢ (𝑥 = 𝐵 → (∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐵} ∈ V))
11	10	elrab3 2963	. . . . 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 7382	. . 3 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
16	15	dmmpt 5232	. 2 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
17	14, 16	eleqtrrdi 2325	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  {crab 2514  Vcvv 2802  ∩ cint 3928   class class class wbr 4088  Oncon0 4460  dom cdm 4725   ≈ cen 6906  cardccrd 7380

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-en 6909  df-card 7382

This theorem is referenced by:  finnum  7386  onenon  7387