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Theorem iscard2 8753
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 8721 . . 3 (card‘𝐴) ∈ On
2 eleq1 2686 . . 3 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 223 . 2 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 cardonle 8734 . . . . . 6 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
54biantrurd 529 . . . . 5 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
6 eqss 3602 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
75, 6syl6rbbr 279 . . . 4 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 ⊆ (card‘𝐴)))
8 oncardval 8732 . . . . 5 (𝐴 ∈ On → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
98sseq2d 3617 . . . 4 (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
107, 9bitrd 268 . . 3 (𝐴 ∈ On → ((card‘𝐴) = 𝐴𝐴 {𝑦 ∈ On ∣ 𝑦𝐴}))
11 ssint 4463 . . . 4 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥)
12 breq1 4621 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1312elrab 3350 . . . . . . . 8 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
14 ensymb 7955 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
1514anbi2i 729 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1613, 15bitri 264 . . . . . . 7 (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ (𝑥 ∈ On ∧ 𝐴𝑥))
1716imbi1i 339 . . . . . 6 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥))
18 impexp 462 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
1917, 18bitri 264 . . . . 5 ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → 𝐴𝑥) ↔ (𝑥 ∈ On → (𝐴𝑥𝐴𝑥)))
2019ralbii2 2973 . . . 4 (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦𝐴}𝐴𝑥 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2111, 20bitri 264 . . 3 (𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥))
2210, 21syl6bb 276 . 2 (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
233, 22biadan2 673 1 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  wss 3559   cint 4445   class class class wbr 4618  Oncon0 5687  cfv 5852  cen 7903  cardccrd 8712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-er 7694  df-en 7907  df-card 8716
This theorem is referenced by:  harcard  8755
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