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Theorem cardcl 7182
Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
cardcl (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ (cardβ€˜π΄) ∈ On)
Distinct variable group:   𝑦,𝐴

Proof of Theorem cardcl
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 df-card 7181 . . . 4 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
21a1i 9 . . 3 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯}))
3 breq2 4009 . . . . . 6 (π‘₯ = 𝐴 β†’ (𝑦 β‰ˆ π‘₯ ↔ 𝑦 β‰ˆ 𝐴))
43rabbidv 2728 . . . . 5 (π‘₯ = 𝐴 β†’ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} = {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
54inteqd 3851 . . . 4 (π‘₯ = 𝐴 β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
65adantl 277 . . 3 ((βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 ∧ π‘₯ = 𝐴) β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
7 encv 6748 . . . . 5 (𝑦 β‰ˆ 𝐴 β†’ (𝑦 ∈ V ∧ 𝐴 ∈ V))
87simprd 114 . . . 4 (𝑦 β‰ˆ 𝐴 β†’ 𝐴 ∈ V)
98rexlimivw 2590 . . 3 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ 𝐴 ∈ V)
10 intexrabim 4155 . . 3 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ V)
112, 6, 9, 10fvmptd 5599 . 2 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
12 onintrab2im 4519 . 2 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ On)
1311, 12eqeltrd 2254 1 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ (cardβ€˜π΄) ∈ On)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456  {crab 2459  Vcvv 2739  βˆ© cint 3846   class class class wbr 4005   ↦ cmpt 4066  Oncon0 4365  β€˜cfv 5218   β‰ˆ cen 6740  cardccrd 7180
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
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 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-en 6743  df-card 7181
This theorem is referenced by: (None)
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