Theorem cardcl 7287

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 7285	. . . 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 4047	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
4	3	rabbidv 2760	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
5	4	inteqd 3889	. . . 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 6832	. . . . 5 ⊢ (𝑦 ≈ 𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
8	7	simprd 114	. . . 4 ⊢ (𝑦 ≈ 𝐴 → 𝐴 ∈ V)
9	8	rexlimivw 2618	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝐴 ∈ V)
10		intexrabim 4196	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
11	2, 6, 9, 10	fvmptd 5659	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
12		onintrab2im 4565	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ On)
13	11, 12	eqeltrd 2281	1 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)

Syntax hints:   → wi 4   = wceq 1372   ∈ wcel 2175  ∃wrex 2484  {crab 2487  Vcvv 2771  ∩ cint 3884   class class class wbr 4043   ↦ cmpt 4104  Oncon0 4409  ‘cfv 5270   ≈ cen 6824  cardccrd 7283

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-en 6827  df-card 7285

This theorem is referenced by:  ficardon  7295