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Theorem isnumi 6907
 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 3870 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
21rspcev 2736 . . . 4 ((𝐴 ∈ On ∧ 𝐴𝐵) → ∃𝑦 ∈ On 𝑦𝐵)
3 intexrabim 4010 . . . 4 (∃𝑦 ∈ On 𝑦𝐵 {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
42, 3syl 14 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
5 encv 6543 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65simprd 113 . . . . 5 (𝐴𝐵𝐵 ∈ V)
7 breq2 3871 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
87rabbidv 2622 . . . . . . . 8 (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
98inteqd 3715 . . . . . . 7 (𝑥 = 𝐵 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
109eleq1d 2163 . . . . . 6 (𝑥 = 𝐵 → ( {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1110elrab3 2786 . . . . 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 272 . . 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 6905 . . 3 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
1615dmmpt 4960 . 2 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
1714, 16syl6eleqr 2188 1 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∃wrex 2371  {crab 2374  Vcvv 2633  ∩ cint 3710   class class class wbr 3867  Oncon0 4214  dom cdm 4467   ≈ cen 6535  cardccrd 6904 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-xp 4473  df-rel 4474  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-en 6538  df-card 6905 This theorem is referenced by:  finnum  6908  onenon  6909
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