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Theorem iscard2 9912
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 9880 . . 3 (card‘𝐴) ∈ On
2 eleq1 2825 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eqss 3959 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
5 cardonle 9893 . . . . . 6 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
65biantrurd 533 . . . . 5 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
74, 6bitr4id 289 . . . 4 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 ⊆ (card‘𝐴)))
8 oncardval 9891 . . . . 5 (𝐴 ∈ On → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
98sseq2d 3976 . . . 4 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
107, 9bitrd 278 . . 3 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
11 ssint 4925 . . . 4 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥)
12 breq1 5108 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1312elrab 3645 . . . . . . . 8 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
14 ensymb 8942 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
1514anbi2i 623 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1613, 15bitri 274 . . . . . . 7 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1716imbi1i 349 . . . . . 6 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥))
18 impexp 451 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
1917, 18bitri 274 . . . . 5 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
2019ralbii2 3092 . . . 4 (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2111, 20bitri 274 . . 3 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2210, 21bitrdi 286 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
233, 22biadanii 820 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wss 3910   cint 4907   class class class wbr 5105  Oncon0 6317  cfv 6496  cen 8880  cardccrd 9871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-er 8648  df-en 8884  df-card 9875
This theorem is referenced by:  harcard  9914
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