Theorem oncard 3636

Description: A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85.

Assertion

Ref	Expression
oncard	⊢ (∃x A = (card ‘x) ↔ A = (card ‘A))

Distinct variable group(s):   x,A

Proof of Theorem oncard

Step	Hyp	Ref	Expression
1		cardid 3635	. . . . . . 7 ⊢ (card ‘x) ≈ x
2		breq1 2065	. . . . . . 7 ⊢ (A = (card ‘x) → (A ≈ x ↔ (card ‘x) ≈ x))
3	1, 2	mpbiri 169	. . . . . 6 ⊢ (A = (card ‘x) → A ≈ x)
4		cardid 3635	. . . . . . 7 ⊢ (card ‘A) ≈ A
5		entrt 3319	. . . . . . 7 ⊢ (((card ‘A) ≈ A ∧ A ≈ x) → (card ‘A) ≈ x)
6	4, 5	mpan 518	. . . . . 6 ⊢ (A ≈ x → (card ‘A) ≈ x)
7		cardon 3634	. . . . . . . 8 ⊢ (card ‘A) ∈ On
8		breq1 2065	. . . . . . . . 9 ⊢ (y = (card ‘A) → (y ≈ x ↔ (card ‘A) ≈ x))
9	8	onintss 2266	. . . . . . . 8 ⊢ ((card ‘A) ∈ On → ((card ‘A) ≈ x → ∩{y ∈ On∣y ≈ x} ⊆ (card ‘A)))
10	7, 9	ax-mp 6	. . . . . . 7 ⊢ ((card ‘A) ≈ x → ∩{y ∈ On∣y ≈ x} ⊆ (card ‘A))
11		cardval 3633	. . . . . . 7 ⊢ (card ‘x) = ∩{y ∈ On∣y ≈ x}
12	10, 11	syl5ss 1544	. . . . . 6 ⊢ ((card ‘A) ≈ x → (card ‘x) ⊆ (card ‘A))
13	3, 6, 12	3syl 21	. . . . 5 ⊢ (A = (card ‘x) → (card ‘x) ⊆ (card ‘A))
14		sseq1 1521	. . . . 5 ⊢ (A = (card ‘x) → (A ⊆ (card ‘A) ↔ (card ‘x) ⊆ (card ‘A)))
15	13, 14	mpbird 171	. . . 4 ⊢ (A = (card ‘x) → A ⊆ (card ‘A))
16		cardon 3634	. . . . . 6 ⊢ (card ‘x) ∈ On
17		eleq1 1149	. . . . . 6 ⊢ (A = (card ‘x) → (A ∈ On ↔ (card ‘x) ∈ On))
18	16, 17	mpbiri 169	. . . . 5 ⊢ (A = (card ‘x) → A ∈ On)
19		cardonle 3629	. . . . 5 ⊢ (A ∈ On → (card ‘A) ⊆ A)
20	18, 19	syl 12	. . . 4 ⊢ (A = (card ‘x) → (card ‘A) ⊆ A)
21	15, 20	eqssd 1518	. . 3 ⊢ (A = (card ‘x) → A = (card ‘A))
22	21	19.23aiv 952	. 2 ⊢ (∃x A = (card ‘x) → A = (card ‘A))
23		fvex 2838	. . . 4 ⊢ (card ‘A) ∈ V
24		eleq1 1149	. . . 4 ⊢ (A = (card ‘A) → (A ∈ V ↔ (card ‘A) ∈ V))
25	23, 24	mpbiri 169	. . 3 ⊢ (A = (card ‘A) → A ∈ V)
26		fveq2 2832	. . . . 5 ⊢ (x = A → (card ‘x) = (card ‘A))
27	26	cleq2d 1112	. . . 4 ⊢ (x = A → (A = (card ‘x) ↔ A = (card ‘A)))
28	27	cla4egv 1397	. . 3 ⊢ (A ∈ V → (A = (card ‘A) → ∃x A = (card ‘x)))
29	25, 28	mpcom 49	. 2 ⊢ (A = (card ‘A) → ∃x A = (card ‘x))
30	22, 29	impbi 139	1 ⊢ (∃x A = (card ‘x) ↔ A = (card ‘A))

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∩cint 1965   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≈ cen 3271  cardccrd 3620

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-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-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-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  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-suc 2205  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-er 3200  df-en 3274  df-card 3623

metamath.org