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Theorem isnumi 7043
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 3932 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
21rspcev 2789 . . . 4 ((𝐴 ∈ On ∧ 𝐴𝐵) → ∃𝑦 ∈ On 𝑦𝐵)
3 intexrabim 4078 . . . 4 (∃𝑦 ∈ On 𝑦𝐵 {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
42, 3syl 14 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
5 encv 6640 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65simprd 113 . . . . 5 (𝐴𝐵𝐵 ∈ V)
7 breq2 3933 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
87rabbidv 2675 . . . . . . . 8 (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
98inteqd 3776 . . . . . . 7 (𝑥 = 𝐵 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
109eleq1d 2208 . . . . . 6 (𝑥 = 𝐵 → ( {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1110elrab3 2841 . . . . 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 275 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → (𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
144, 13mpbird 166 . 2 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V})
15 df-card 7041 . . 3 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
1615dmmpt 5034 . 2 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
1714, 16eleqtrrdi 2233 1 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wrex 2417  {crab 2420  Vcvv 2686   cint 3771   class class class wbr 3929  Oncon0 4285  dom cdm 4539  cen 6632  cardccrd 7040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-en 6635  df-card 7041
This theorem is referenced by:  finnum  7044  onenon  7045
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