Theorem iscard2 9929

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 9897	. . 3 ⊢ (card‘𝐴) ∈ On
2		eleq1 2816	. . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 233	. 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		eqss 3962	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
5		cardonle 9910	. . . . . 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 290	. . . 4 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ (card‘𝐴)))
8		oncardval 9908	. . . . 5 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	8	sseq2d 3979	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
10	7, 9	bitrd 279	. . 3 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
11		ssint 4928	. . . 4 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥)
12		breq1 5110	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
13	12	elrab 3659	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴))
14		ensymb 8973	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)
15	14	anbi2i 623	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
16	13, 15	bitri 275	. . . . . . 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 275	. . . . 5 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
20	19	ralbii2 3071	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
21	11, 20	bitri 275	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
22	10, 21	bitrdi 287	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
23	3, 22	biadanii 821	1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405   ⊆ wss 3914  ∩ cint 4910   class class class wbr 5107  Oncon0 6332  ‘cfv 6511   ≈ cen 8915  cardccrd 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-er 8671  df-en 8919  df-card 9892

This theorem is referenced by:  harcard  9931