Theorem iscard2 9891

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 9859	. . 3 ⊢ (card‘𝐴) ∈ On
2		eleq1 2827	. . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 234	. 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		eqss 3930	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
5		cardonle 9872	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
6	5	biantrurd 537	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
7	4, 6	bitr4id 291	. . . 4 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ (card‘𝐴)))
8		oncardval 9870	. . . . 5 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	8	sseq2d 3947	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
10	7, 9	bitrd 280	. . 3 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ 𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}))
11		ssint 4894	. . . 4 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥)
12		breq1 5075	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
13	12	elrab 3629	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴))
14		ensymb 8939	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)
15	14	anbi2i 629	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
16	13, 15	bitri 276	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥))
17	16	imbi1i 350	. . . . . 6 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → 𝐴 ⊆ 𝑥))
18		impexp 451	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
19	17, 18	bitri 276	. . . . 5 ⊢ ((𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ On → (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
20	19	ralbii2 3081	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
21	11, 20	bitri 276	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))
22	10, 21	bitrdi 288	. 2 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
23	3, 22	biadanii 827	1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391   ⊆ wss 3883  ∩ cint 4877   class class class wbr 5072  Oncon0 6310  ‘cfv 6485   ≈ cen 8880  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-er 8633  df-en 8884  df-card 9854

This theorem is referenced by:  harcard  9893