ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isnumi GIF version

Theorem isnumi 7006
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 3902 . . . . 5 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
21rspcev 2763 . . . 4 ((𝐴 ∈ On ∧ 𝐴𝐵) → ∃𝑦 ∈ On 𝑦𝐵)
3 intexrabim 4048 . . . 4 (∃𝑦 ∈ On 𝑦𝐵 {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
42, 3syl 14 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐵) → {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V)
5 encv 6608 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
65simprd 113 . . . . 5 (𝐴𝐵𝐵 ∈ V)
7 breq2 3903 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
87rabbidv 2649 . . . . . . . 8 (𝑥 = 𝐵 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
98inteqd 3746 . . . . . . 7 (𝑥 = 𝐵 {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝐵})
109eleq1d 2186 . . . . . 6 (𝑥 = 𝐵 → ( {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V ↔ {𝑦 ∈ On ∣ 𝑦𝐵} ∈ V))
1110elrab3 2814 . . . . 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 7004 . . 3 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
1615dmmpt 5004 . 2 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
1714, 16eleqtrrdi 2211 1 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wrex 2394  {crab 2397  Vcvv 2660   cint 3741   class class class wbr 3899  Oncon0 4255  dom cdm 4509  cen 6600  cardccrd 7003
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-en 6603  df-card 7004
This theorem is referenced by:  finnum  7007  onenon  7008
  Copyright terms: Public domain W3C validator