MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscard2 Structured version   Visualization version   GIF version

Theorem iscard2 10007
Description: Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
iscard2 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem iscard2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardon 9975 . . 3 (cardβ€˜π΄) ∈ On
2 eleq1 2817 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . 2 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 eqss 3997 . . . . 5 ((cardβ€˜π΄) = 𝐴 ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄)))
5 cardonle 9988 . . . . . 6 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
65biantrurd 531 . . . . 5 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄))))
74, 6bitr4id 289 . . . 4 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 βŠ† (cardβ€˜π΄)))
8 oncardval 9986 . . . . 5 (𝐴 ∈ On β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
98sseq2d 4014 . . . 4 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ 𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}))
107, 9bitrd 278 . . 3 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}))
11 ssint 4971 . . . 4 (𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ βˆ€π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}𝐴 βŠ† π‘₯)
12 breq1 5155 . . . . . . . . 9 (𝑦 = π‘₯ β†’ (𝑦 β‰ˆ 𝐴 ↔ π‘₯ β‰ˆ 𝐴))
1312elrab 3684 . . . . . . . 8 (π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ (π‘₯ ∈ On ∧ π‘₯ β‰ˆ 𝐴))
14 ensymb 9029 . . . . . . . . 9 (π‘₯ β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ π‘₯)
1514anbi2i 621 . . . . . . . 8 ((π‘₯ ∈ On ∧ π‘₯ β‰ˆ 𝐴) ↔ (π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯))
1613, 15bitri 274 . . . . . . 7 (π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ (π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯))
1716imbi1i 348 . . . . . 6 ((π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ 𝐴 βŠ† π‘₯) ↔ ((π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯) β†’ 𝐴 βŠ† π‘₯))
18 impexp 449 . . . . . 6 (((π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯) β†’ 𝐴 βŠ† π‘₯) ↔ (π‘₯ ∈ On β†’ (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
1917, 18bitri 274 . . . . 5 ((π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ 𝐴 βŠ† π‘₯) ↔ (π‘₯ ∈ On β†’ (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
2019ralbii2 3086 . . . 4 (βˆ€π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}𝐴 βŠ† π‘₯ ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯))
2111, 20bitri 274 . . 3 (𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯))
2210, 21bitrdi 286 . 2 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
233, 22biadanii 820 1 ((cardβ€˜π΄) = 𝐴 ↔ (𝐴 ∈ On ∧ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   βŠ† wss 3949  βˆ© cint 4953   class class class wbr 5152  Oncon0 6374  β€˜cfv 6553   β‰ˆ cen 8967  cardccrd 9966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-er 8731  df-en 8971  df-card 9970
This theorem is referenced by:  harcard  10009
  Copyright terms: Public domain W3C validator