Theorem cardalephex 3691

Description: Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse.

Assertion

Ref	Expression
cardalephex	⊢ (ω ⊆ A → ((card ‘A) = A ↔ ∃x ∈ On A = (ℵ ‘x)))

Distinct variable group(s):   x,A

Proof of Theorem cardalephex

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . . . 6 ⊢ (x = ∩{y ∈ On∣A ⊆ (ℵ ‘y)} → (ℵ ‘x) = (ℵ ‘∩{y ∈ On∣A ⊆ (ℵ ‘y)}))
2	1	cleq2d 1112	. . . . 5 ⊢ (x = ∩{y ∈ On∣A ⊆ (ℵ ‘y)} → (A = (ℵ ‘x) ↔ A = (ℵ ‘∩{y ∈ On∣A ⊆ (ℵ ‘y)})))
3	2	rcla4ev 1403	. . . 4 ⊢ ((∩{y ∈ On∣A ⊆ (ℵ ‘y)} ∈ On ∧ A = (ℵ ‘∩{y ∈ On∣A ⊆ (ℵ ‘y)})) → ∃x ∈ On A = (ℵ ‘x))
4		pm3.26 256	. . . . 5 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → ω ⊆ A)
5		cardaleph 3690	. . . . . . 7 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → A = (ℵ ‘∩{y ∈ On∣A ⊆ (ℵ ‘y)}))
6	5	sseq2d 1528	. . . . . 6 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → (ω ⊆ A ↔ ω ⊆ (ℵ ‘∩{y ∈ On∣A ⊆ (ℵ ‘y)})))
7		alephgeom 3687	. . . . . 6 ⊢ (∩{y ∈ On∣A ⊆ (ℵ ‘y)} ∈ On ↔ ω ⊆ (ℵ ‘∩{y ∈ On∣A ⊆ (ℵ ‘y)}))
8	6, 7	syl6bbr 416	. . . . 5 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → (ω ⊆ A ↔ ∩{y ∈ On∣A ⊆ (ℵ ‘y)} ∈ On))
9	4, 8	mpbid 170	. . . 4 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → ∩{y ∈ On∣A ⊆ (ℵ ‘y)} ∈ On)
10	3, 9, 5	sylanc 361	. . 3 ⊢ ((ω ⊆ A ∧ (card ‘A) = A) → ∃x ∈ On A = (ℵ ‘x))
11	10	exp 291	. 2 ⊢ (ω ⊆ A → ((card ‘A) = A → ∃x ∈ On A = (ℵ ‘x)))
12		alephcard 3673	. . . . . 6 ⊢ (card ‘(ℵ ‘x)) = (ℵ ‘x)
13		fveq2 2832	. . . . . . 7 ⊢ (A = (ℵ ‘x) → (card ‘A) = (card ‘(ℵ ‘x)))
14		id 9	. . . . . . 7 ⊢ (A = (ℵ ‘x) → A = (ℵ ‘x))
15	13, 14	cleq12d 1115	. . . . . 6 ⊢ (A = (ℵ ‘x) → ((card ‘A) = A ↔ (card ‘(ℵ ‘x)) = (ℵ ‘x)))
16	12, 15	mpbiri 169	. . . . 5 ⊢ (A = (ℵ ‘x) → (card ‘A) = A)
17	16	a1i 7	. . . 4 ⊢ (x ∈ On → (A = (ℵ ‘x) → (card ‘A) = A))
18	17	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ On A = (ℵ ‘x) → (card ‘A) = A)
19	18	a1i 7	. 2 ⊢ (ω ⊆ A → (∃x ∈ On A = (ℵ ‘x) → (card ‘A) = A))
20	11, 19	impbid 397	1 ⊢ (ω ⊆ A → ((card ‘A) = A ↔ ∃x ∈ On A = (ℵ ‘x)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204   ⊆ wss 1487  ∩cint 1965  Oncon0 2199  ωcom 2372   ‘cfv 2422  cardccrd 3620  ℵcale 3621

This theorem is referenced by:  isinfcard 3692

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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