Theorem cardcl 7037

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 7036	. . . 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 3933	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
4	3	rabbidv 2675	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
5	4	inteqd 3776	. . . 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 6640	. . . . 5 ⊢ (𝑦 ≈ 𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
8	7	simprd 113	. . . 4 ⊢ (𝑦 ≈ 𝐴 → 𝐴 ∈ V)
9	8	rexlimivw 2545	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝐴 ∈ V)
10		intexrabim 4078	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
11	2, 6, 9, 10	fvmptd 5502	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
12		onintrab2im 4434	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ On)
13	11, 12	eqeltrd 2216	1 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  {crab 2420  Vcvv 2686  ∩ cint 3771   class class class wbr 3929   ↦ cmpt 3989  Oncon0 4285  ‘cfv 5123   ≈ cen 6632  cardccrd 7035

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-en 6635  df-card 7036

This theorem is referenced by: (None)