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Theorem isnumi 6713
Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
isnumi ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)

Proof of Theorem isnumi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3814 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
21rspcev 2712 . . . 4 ((𝐴 ∈ On ∧ 𝐴𝐵) → ∃𝑦 ∈ On 𝑦𝐵)
3 intexrabim 3954 . . . 4 (∃𝑦 ∈ On 𝑦𝐵 {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
42, 3syl 14 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
5 encv 6393 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65simprd 112 . . . . 5 (𝐴𝐵𝐵 ∈ V)
7 breq2 3815 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
87rabbidv 2601 . . . . . . . 8 (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
98inteqd 3667 . . . . . . 7 (𝑥 = 𝐵 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
109eleq1d 2151 . . . . . 6 (𝑥 = 𝐵 → ( {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1110elrab3 2760 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
126, 11syl 14 . . . 4 (𝐴𝐵 → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1312adantl 271 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
144, 13mpbird 165 . 2 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V})
15 df-card 6711 . . 3 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
1615dmmpt 4880 . 2 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
1714, 16syl6eleqr 2176 1 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wrex 2354  {crab 2357  Vcvv 2612   cint 3662   class class class wbr 3811  Oncon0 4154  dom cdm 4401  cen 6385  cardccrd 6710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-xp 4407  df-rel 4408  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-en 6388  df-card 6711
This theorem is referenced by:  finnum  6714  onenon  6715
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