Theorem iscard2 9981

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 9949	. . 3 ⊢ (card‘𝐴) ∈ On
2		eleq1 2848	. . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 236	. 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		eqss 3946	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
5		cardonle 9962	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
6	5	biantrurd 542	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
7	4, 6	bitr4id 293	. . . 4 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ (card‘𝐴)))
8		oncardval 9960	. . . . 5 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	8	sseq2d 3963	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
10	7, 9	bitrd 282	. . 3 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
11		ssint 4924	. . . 4 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥)
12		breq1 5106	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
13	12	elrab 3645	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴))
14		ensymb 9008	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)
15	14	anbi2i 635	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
16	13, 15	bitri 278	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
17	16	imbi1i 352	. . . . . 6 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → 𝐴 ⊆ 𝑥))
18		impexp 456	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
19	17, 18	bitri 278	. . . . 5 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
20	19	ralbii2 3104	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
21	11, 20	bitri 278	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
22	10, 21	bitrdi 290	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
23	3, 22	biadanii 834	1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ⊆ wss 3899  ∩ cint 4907   class class class wbr 5103  Oncon0 6357  ‘cfv 6533   ≈ cen 8949  cardccrd 9940

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-er 8696  df-en 8953  df-card 9944

This theorem is used by:  harcard  9983