Theorem iscard2 9665

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 (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem iscard2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardon 9633	. . 3 ⊢ (card‘𝐴) ∈ On
2		eleq1 2826	. . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 232	. 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		eqss 3932	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
5		cardonle 9646	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
6	5	biantrurd 532	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
7	4, 6	bitr4id 289	. . . 4 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ (card‘𝐴)))
8		oncardval 9644	. . . . 5 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	8	sseq2d 3949	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
10	7, 9	bitrd 278	. . 3 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
11		ssint 4892	. . . 4 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥)
12		breq1 5073	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
13	12	elrab 3617	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴))
14		ensymb 8743	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)
15	14	anbi2i 622	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
16	13, 15	bitri 274	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
17	16	imbi1i 349	. . . . . 6 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → 𝐴 ⊆ 𝑥))
18		impexp 450	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
19	17, 18	bitri 274	. . . . 5 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
20	19	ralbii2 3088	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
21	11, 20	bitri 274	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
22	10, 21	bitrdi 286	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
23	3, 22	biadanii 818	1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  ∩ cint 4876   class class class wbr 5070  Oncon0 6251  ‘cfv 6418   ≈ cen 8688  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-er 8456  df-en 8692  df-card 9628

This theorem is referenced by:  harcard  9667