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Theorem cardcl 7179
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 7178 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21a1i 9 . . 3 (∃𝑦 ∈ On 𝑦𝐴 → card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
3 breq2 4007 . . . . . 6 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
43rabbidv 2726 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
54inteqd 3849 . . . 4 (𝑥 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 277 . . 3 ((∃𝑦 ∈ On 𝑦𝐴𝑥 = 𝐴) → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
7 encv 6745 . . . . 5 (𝑦𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
87simprd 114 . . . 4 (𝑦𝐴𝐴 ∈ V)
98rexlimivw 2590 . . 3 (∃𝑦 ∈ On 𝑦𝐴𝐴 ∈ V)
10 intexrabim 4153 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
112, 6, 9, 10fvmptd 5597 . 2 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
12 onintrab2im 4517 . 2 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ On)
1311, 12eqeltrd 2254 1 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wrex 2456  {crab 2459  Vcvv 2737   cint 3844   class class class wbr 4003  cmpt 4064  Oncon0 4363  cfv 5216  cen 6737  cardccrd 7177
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-en 6740  df-card 7178
This theorem is referenced by: (None)
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