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Theorem cardcl 6318
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 6317 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21a1i 9 . . 3 (∃𝑦 ∈ On 𝑦𝐴 → card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
3 breq2 3765 . . . . . 6 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
43rabbidv 2546 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
54inteqd 3617 . . . 4 (𝑥 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 262 . . 3 ((∃𝑦 ∈ On 𝑦𝐴𝑥 = 𝐴) → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
7 encv 6190 . . . . 5 (𝑦𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
87simprd 107 . . . 4 (𝑦𝐴𝐴 ∈ V)
98rexlimivw 2426 . . 3 (∃𝑦 ∈ On 𝑦𝐴𝐴 ∈ V)
10 intexrabim 3904 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
112, 6, 9, 10fvmptd 5216 . 2 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
12 onintrab2im 4216 . 2 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ On)
1311, 12eqeltrd 2114 1 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  wrex 2304  {crab 2307  Vcvv 2554   cint 3612   class class class wbr 3761  cmpt 3815  Oncon0 4072  cfv 4865  cen 6182  cardccrd 6316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-id 4027  df-iord 4075  df-on 4077  df-suc 4080  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-iota 4830  df-fun 4867  df-fv 4873  df-en 6185  df-card 6317
This theorem is referenced by: (None)
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