Theorem cardcl 7137

Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
cardcl	⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)

Distinct variable group:   𝑦,𝐴

Proof of Theorem cardcl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-card 7136	. . . 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 3986	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
4	3	rabbidv 2715	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
5	4	inteqd 3829	. . . 4 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
6	5	adantl 275	. . 3 ⊢ ((∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
7		encv 6712	. . . . 5 ⊢ (𝑦 ≈ 𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
8	7	simprd 113	. . . 4 ⊢ (𝑦 ≈ 𝐴 → 𝐴 ∈ V)
9	8	rexlimivw 2579	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝐴 ∈ V)
10		intexrabim 4132	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
11	2, 6, 9, 10	fvmptd 5567	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
12		onintrab2im 4495	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ On)
13	11, 12	eqeltrd 2243	1 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  {crab 2448  Vcvv 2726  ∩ cint 3824   class class class wbr 3982   ↦ cmpt 4043  Oncon0 4341  ‘cfv 5188   ≈ cen 6704  cardccrd 7135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-en 6707  df-card 7136

This theorem is referenced by: (None)