Theorem iscard3 9780

Description: Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)

Assertion

Ref	Expression
iscard3	⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ))

Proof of Theorem iscard3

Step	Hyp	Ref	Expression
1		cardon 9633	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
2		eleq1 2826	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 232	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		eloni 6261	. . . . . . . 8 ⊢ (𝐴 ∈ On → Ord 𝐴)
5	3, 4	syl 17	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → Ord 𝐴)
6		ordom 7697	. . . . . . 7 ⊢ Ord ω
7		ordtri2or 6346	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
8	5, 6, 7	sylancl 585	. . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴))
9	8	ord 860	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴))
10		isinfcard 9779	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
11	10	biimpi 215	. . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ)
12	11	expcom 413	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 → 𝐴 ∈ ran ℵ))
13	9, 12	syld 47	. . . 4 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ))
14	13	orrd 859	. . 3 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
15		cardnn 9652	. . . 4 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
16	10	bicomi 223	. . . . 5 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
17	16	simprbi 496	. . . 4 ⊢ (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴)
18	15, 17	jaoi 853	. . 3 ⊢ ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴)
19	14, 18	impbii 208	. 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
20		elun 4079	. 2 ⊢ (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ))
21	19, 20	bitr4i 277	1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ∪ cun 3881   ⊆ wss 3883  ran crn 5581  Ord word 6250  Oncon0 6251  ‘cfv 6418  ωcom 7687  cardccrd 9624  ℵcale 9625

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-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

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-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  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-iun 4923  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-se 5536  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-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  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-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629

This theorem is referenced by:  cardnum  9781  carduniima  9783  cardinfima  9784  cfpwsdom  10271  gch2  10362