Theorem iscard2 9958

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 9926	. . 3 ⊢ (card‘𝐴) ∈ On
2		eleq1 2851	. . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 236	. 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		eqss 3952	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
5		cardonle 9939	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
6	5	biantrurd 541	. . . . 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 9937	. . . . 5 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	8	sseq2d 3969	. . . 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 4929	. . . 4 ⊢ (𝐴 ⊆ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ∀𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}𝐴 ⊆ 𝑥)
12		breq1 5112	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
13	12	elrab 3650	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴))
14		ensymb 8995	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)
15	14	anbi2i 634	. . . . . . . 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 455	. . . . . 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 3107	. . . 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 833	1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ⊆ wss 3905  ∩ cint 4912   class class class wbr 5109  Oncon0 6360  ‘cfv 6536   ≈ cen 8936  cardccrd 9917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8690  df-en 8940  df-card 9921

This theorem is referenced by:  harcard  9960