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Theorem cardcl 6948
Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
cardcl (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)
Distinct variable group:   𝑦,𝐴

Proof of Theorem cardcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-card 6947 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21a1i 9 . . 3 (∃𝑦 ∈ On 𝑦𝐴 → card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
3 breq2 3879 . . . . . 6 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
43rabbidv 2630 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
54inteqd 3723 . . . 4 (𝑥 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 273 . . 3 ((∃𝑦 ∈ On 𝑦𝐴𝑥 = 𝐴) → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
7 encv 6570 . . . . 5 (𝑦𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
87simprd 113 . . . 4 (𝑦𝐴𝐴 ∈ V)
98rexlimivw 2504 . . 3 (∃𝑦 ∈ On 𝑦𝐴𝐴 ∈ V)
10 intexrabim 4018 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
112, 6, 9, 10fvmptd 5434 . 2 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
12 onintrab2im 4372 . 2 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ On)
1311, 12eqeltrd 2176 1 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  wrex 2376  {crab 2379  Vcvv 2641   cint 3718   class class class wbr 3875  cmpt 3929  Oncon0 4223  cfv 5059  cen 6562  cardccrd 6946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-en 6565  df-card 6947
This theorem is referenced by: (None)
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