Theorem cardcl 7288

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 7286	. . . 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 4048	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
4	3	rabbidv 2761	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
5	4	inteqd 3890	. . . 4 ⊢ (𝑥 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
6	5	adantl 277	. . 3 ⊢ ((∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑥 = 𝐴) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
7		encv 6833	. . . . 5 ⊢ (𝑦 ≈ 𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
8	7	simprd 114	. . . 4 ⊢ (𝑦 ≈ 𝐴 → 𝐴 ∈ V)
9	8	rexlimivw 2619	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝐴 ∈ V)
10		intexrabim 4197	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
11	2, 6, 9, 10	fvmptd 5660	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
12		onintrab2im 4566	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ On)
13	11, 12	eqeltrd 2282	1 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2176  ∃wrex 2485  {crab 2488  Vcvv 2772  ∩ cint 3885   class class class wbr 4044   ↦ cmpt 4105  Oncon0 4410  ‘cfv 5271   ≈ cen 6825  cardccrd 7284

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-en 6828  df-card 7286

This theorem is referenced by:  ficardon  7296