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Theorem iscard2 10014
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 9982 . . 3 (card‘𝐴) ∈ On
2 eleq1 2827 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 233 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 eqss 4011 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
5 cardonle 9995 . . . . . 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 9993 . . . . 5 (𝐴 ∈ On → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
98sseq2d 4028 . . . 4 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
107, 9bitrd 279 . . 3 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
11 ssint 4969 . . . 4 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥)
12 breq1 5151 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1312elrab 3695 . . . . . . . 8 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
14 ensymb 9041 . . . . . . . . 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 3087 . . . 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 822 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963   cint 4951   class class class wbr 5148  Oncon0 6386  cfv 6563  cen 8981  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-card 9977
This theorem is referenced by:  harcard  10016
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