Theorem iscard3 3693

Description: Two ways of expressing the property of being a cardinal number.

Assertion

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

Proof of Theorem iscard3

Step	Hyp	Ref	Expression
1		cardon 3634	. . . . . . . 8 ⊢ (card ‘A) ∈ On
2		eleq1 1149	. . . . . . . 8 ⊢ ((card ‘A) = A → ((card ‘A) ∈ On ↔ A ∈ On))
3	1, 2	mpbii 168	. . . . . . 7 ⊢ ((card ‘A) = A → A ∈ On)
4		eloni 2209	. . . . . . 7 ⊢ (A ∈ On → Ord A)
5		ordom 2382	. . . . . . . 8 ⊢ Ord ω
6		ordtri2or 2328	. . . . . . . 8 ⊢ ((Ord A ∧ Ord ω) → (A ∈ ω ∨ ω ⊆ A))
7	5, 6	mpan2 519	. . . . . . 7 ⊢ (Ord A → (A ∈ ω ∨ ω ⊆ A))
8	3, 4, 7	3syl 21	. . . . . 6 ⊢ ((card ‘A) = A → (A ∈ ω ∨ ω ⊆ A))
9	8	ord 202	. . . . 5 ⊢ ((card ‘A) = A → (¬ A ∈ ω → ω ⊆ A))
10		isinfcard 3692	. . . . . . . 8 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) ↔ A ∈ ran ℵ)
11	10	biimp 133	. . . . . . 7 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → A ∈ ran ℵ)
12	11	exp 291	. . . . . 6 ⊢ (ω ⊆ A → ((card ‘A) = A → A ∈ ran ℵ))
13	12	com12 13	. . . . 5 ⊢ ((card ‘A) = A → (ω ⊆ A → A ∈ ran ℵ))
14	9, 13	syld 27	. . . 4 ⊢ ((card ‘A) = A → (¬ A ∈ ω → A ∈ ran ℵ))
15	14	orrd 203	. . 3 ⊢ ((card ‘A) = A → (A ∈ ω ∨ A ∈ ran ℵ))
16		cardnn 3631	. . . 4 ⊢ (A ∈ ω → (card ‘A) = A)
17	10	bicomi 150	. . . . 5 ⊢ (A ∈ ran ℵ ↔ (ω ⊆ A ∧ (card ‘A) = A))
18	17	pm3.27bd 263	. . . 4 ⊢ (A ∈ ran ℵ → (card ‘A) = A)
19	16, 18	jaoi 275	. . 3 ⊢ ((A ∈ ω ∨ A ∈ ran ℵ) → (card ‘A) = A)
20	15, 19	impbi 139	. 2 ⊢ ((card ‘A) = A ↔ (A ∈ ω ∨ A ∈ ran ℵ))
21		elun 1601	. 2 ⊢ (A ∈ (ω ∪ ran ℵ) ↔ (A ∈ ω ∨ A ∈ ran ℵ))
22	20, 21	bitr4 154	1 ⊢ ((card ‘A) = A ↔ A ∈ (ω ∪ ran ℵ))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487  Ord word 2198  Oncon0 2199  ωcom 2372  ran crn 2411   ‘cfv 2422  cardccrd 3620  ℵcale 3621

This theorem is referenced by:  cardnum 3694  carduniima 3695  cardinfima 3696  elcard 3713

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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