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Theorem iscard2 9970
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 9938 . . 3 (cardβ€˜π΄) ∈ On
2 eleq1 2815 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . 2 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 eqss 3992 . . . . 5 ((cardβ€˜π΄) = 𝐴 ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄)))
5 cardonle 9951 . . . . . 6 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
65biantrurd 532 . . . . 5 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ ((cardβ€˜π΄) βŠ† 𝐴 ∧ 𝐴 βŠ† (cardβ€˜π΄))))
74, 6bitr4id 290 . . . 4 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 βŠ† (cardβ€˜π΄)))
8 oncardval 9949 . . . . 5 (𝐴 ∈ On β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
98sseq2d 4009 . . . 4 (𝐴 ∈ On β†’ (𝐴 βŠ† (cardβ€˜π΄) ↔ 𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}))
107, 9bitrd 279 . . 3 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}))
11 ssint 4961 . . . 4 (𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ βˆ€π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴}𝐴 βŠ† π‘₯)
12 breq1 5144 . . . . . . . . 9 (𝑦 = π‘₯ β†’ (𝑦 β‰ˆ 𝐴 ↔ π‘₯ β‰ˆ 𝐴))
1312elrab 3678 . . . . . . . 8 (π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ (π‘₯ ∈ On ∧ π‘₯ β‰ˆ 𝐴))
14 ensymb 8997 . . . . . . . . 9 (π‘₯ β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ π‘₯)
1514anbi2i 622 . . . . . . . 8 ((π‘₯ ∈ On ∧ π‘₯ β‰ˆ 𝐴) ↔ (π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯))
1613, 15bitri 275 . . . . . . 7 (π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ (π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯))
1716imbi1i 349 . . . . . 6 ((π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ 𝐴 βŠ† π‘₯) ↔ ((π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯) β†’ 𝐴 βŠ† π‘₯))
18 impexp 450 . . . . . 6 (((π‘₯ ∈ On ∧ 𝐴 β‰ˆ π‘₯) β†’ 𝐴 βŠ† π‘₯) ↔ (π‘₯ ∈ On β†’ (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
1917, 18bitri 275 . . . . 5 ((π‘₯ ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ 𝐴 βŠ† π‘₯) ↔ (π‘₯ ∈ On β†’ (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
2019ralbii2 3083 . . . 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 275 . . 3 (𝐴 βŠ† ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯))
2210, 21bitrdi 287 . 2 (𝐴 ∈ On β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆ€π‘₯ ∈ On (𝐴 β‰ˆ π‘₯ β†’ 𝐴 βŠ† π‘₯)))
233, 22biadanii 819 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   βŠ† wss 3943  βˆ© cint 4943   class class class wbr 5141  Oncon0 6357  β€˜cfv 6536   β‰ˆ cen 8935  cardccrd 9929
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8702  df-en 8939  df-card 9933
This theorem is referenced by:  harcard  9972
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