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Theorem cardcl 6419
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 6418 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21a1i 9 . . 3 (∃𝑦 ∈ On 𝑦𝐴 → card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥}))
3 breq2 3796 . . . . . 6 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
43rabbidv 2566 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
54inteqd 3648 . . . 4 (𝑥 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
65adantl 266 . . 3 ((∃𝑦 ∈ On 𝑦𝐴𝑥 = 𝐴) → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐴})
7 encv 6258 . . . . 5 (𝑦𝐴 → (𝑦 ∈ V ∧ 𝐴 ∈ V))
87simprd 111 . . . 4 (𝑦𝐴𝐴 ∈ V)
98rexlimivw 2446 . . 3 (∃𝑦 ∈ On 𝑦𝐴𝐴 ∈ V)
10 intexrabim 3935 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
112, 6, 9, 10fvmptd 5281 . 2 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
12 onintrab2im 4272 . 2 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ On)
1311, 12eqeltrd 2130 1 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  wrex 2324  {crab 2327  Vcvv 2574   cint 3643   class class class wbr 3792  cmpt 3846  Oncon0 4128  cfv 4930  cen 6250  cardccrd 6417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-en 6253  df-card 6418
This theorem is referenced by: (None)
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