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Theorem iscard2 10017
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 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem iscard2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardon 9985 . . 3 (card‘𝐴) ∈ On
2 eleq1 2828 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 233 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eqss 3998 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
5 cardonle 9998 . . . . . 6 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
65biantrurd 532 . . . . 5 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
74, 6bitr4id 290 . . . 4 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 ⊆ (card‘𝐴)))
8 oncardval 9996 . . . . 5 (𝐴 ∈ On → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
98sseq2d 4015 . . . 4 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
107, 9bitrd 279 . . 3 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
11 ssint 4963 . . . 4 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥)
12 breq1 5145 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1312elrab 3691 . . . . . . . 8 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
14 ensymb 9043 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
1514anbi2i 623 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1613, 15bitri 275 . . . . . . 7 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1716imbi1i 349 . . . . . 6 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥))
18 impexp 450 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
1917, 18bitri 275 . . . . 5 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
2019ralbii2 3088 . . . 4 (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2111, 20bitri 275 . . 3 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2210, 21bitrdi 287 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
233, 22biadanii 821 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  wss 3950   cint 4945   class class class wbr 5142  Oncon0 6383  cfv 6560  cen 8983  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-er 8746  df-en 8987  df-card 9980
This theorem is referenced by:  harcard  10019
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