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Theorem cardcl 6359
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 6358 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21a1i 9 . . 3 (∃𝑦 ∈ On 𝑦𝐴 → card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
3 breq2 3768 . . . . . 6 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
43rabbidv 2549 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
54inteqd 3620 . . . 4 (𝑥 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 262 . . 3 ((∃𝑦 ∈ On 𝑦𝐴𝑥 = 𝐴) → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
7 encv 6227 . . . . 5 (𝑦𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
87simprd 107 . . . 4 (𝑦𝐴𝐴 ∈ V)
98rexlimivw 2429 . . 3 (∃𝑦 ∈ On 𝑦𝐴𝐴 ∈ V)
10 intexrabim 3907 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
112, 6, 9, 10fvmptd 5253 . 2 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
12 onintrab2im 4244 . 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 2307  {crab 2310  Vcvv 2557   cint 3615   class class class wbr 3764  cmpt 3818  Oncon0 4100  cfv 4902  cen 6219  cardccrd 6357
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 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
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 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-en 6222  df-card 6358
This theorem is referenced by: (None)
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