Theorem iscard2 3660

Description: Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225.

Assertion

Ref	Expression
iscard2	⊢ ((card ‘A) = A ↔ (A ∈ On ∧ ∀x ∈ On (A ≈ x → A ⊆ x)))

Distinct variable group(s):   x,A

Proof of Theorem iscard2

Step	Hyp	Ref	Expression
1		cardon 3634	. . . 4 ⊢ (card ‘A) ∈ On
2		eleq1 1149	. . . 4 ⊢ ((card ‘A) = A → ((card ‘A) ∈ On ↔ A ∈ On))
3	1, 2	mpbii 168	. . 3 ⊢ ((card ‘A) = A → A ∈ On)
4	3	pm4.71ri 484	. 2 ⊢ ((card ‘A) = A ↔ (A ∈ On ∧ (card ‘A) = A))
5		cardonle 3629	. . . . . 6 ⊢ (A ∈ On → (card ‘A) ⊆ A)
6	5	biantrurd 546	. . . . 5 ⊢ (A ∈ On → (A ⊆ (card ‘A) ↔ ((card ‘A) ⊆ A ∧ A ⊆ (card ‘A))))
7		eqss 1516	. . . . 5 ⊢ ((card ‘A) = A ↔ ((card ‘A) ⊆ A ∧ A ⊆ (card ‘A)))
8	6, 7	syl6rbbr 417	. . . 4 ⊢ (A ∈ On → ((card ‘A) = A ↔ A ⊆ (card ‘A)))
9		ensymg 3316	. . . . . . . . . . . 12 ⊢ (A ∈ On → (x ≈ A → A ≈ x))
10		visset 1350	. . . . . . . . . . . . . 14 ⊢ x ∈ V
11	10	ensym 3317	. . . . . . . . . . . . 13 ⊢ (A ≈ x → x ≈ A)
12	11	a1i 7	. . . . . . . . . . . 12 ⊢ (A ∈ On → (A ≈ x → x ≈ A))
13	9, 12	impbid 397	. . . . . . . . . . 11 ⊢ (A ∈ On → (x ≈ A ↔ A ≈ x))
14	13	anbi2d 468	. . . . . . . . . 10 ⊢ (A ∈ On → ((x ∈ On ∧ x ≈ A) ↔ (x ∈ On ∧ A ≈ x)))
15		breq1 2065	. . . . . . . . . . 11 ⊢ (y = x → (y ≈ A ↔ x ≈ A))
16	15	elrab 1422	. . . . . . . . . 10 ⊢ (x ∈ {y ∈ On∣y ≈ A} ↔ (x ∈ On ∧ x ≈ A))
17	14, 16	syl5bb 410	. . . . . . . . 9 ⊢ (A ∈ On → (x ∈ {y ∈ On∣y ≈ A} ↔ (x ∈ On ∧ A ≈ x)))
18	17	imbi1d 465	. . . . . . . 8 ⊢ (A ∈ On → ((x ∈ {y ∈ On∣y ≈ A} → A ⊆ x) ↔ ((x ∈ On ∧ A ≈ x) → A ⊆ x)))
19		impexp 276	. . . . . . . 8 ⊢ (((x ∈ On ∧ A ≈ x) → A ⊆ x) ↔ (x ∈ On → (A ≈ x → A ⊆ x)))
20	18, 19	syl6bb 414	. . . . . . 7 ⊢ (A ∈ On → ((x ∈ {y ∈ On∣y ≈ A} → A ⊆ x) ↔ (x ∈ On → (A ≈ x → A ⊆ x))))
21	20	biraldv2 1221	. . . . . 6 ⊢ (A ∈ On → (∀x ∈ {y ∈ On∣y ≈ A}A ⊆ x ↔ ∀x ∈ On (A ≈ x → A ⊆ x)))
22		ssint 1980	. . . . . 6 ⊢ (A ⊆ ∩{y ∈ On∣y ≈ A} ↔ ∀x ∈ {y ∈ On∣y ≈ A}A ⊆ x)
23	21, 22	syl5bb 410	. . . . 5 ⊢ (A ∈ On → (A ⊆ ∩{y ∈ On∣y ≈ A} ↔ ∀x ∈ On (A ≈ x → A ⊆ x)))
24		cardval 3633	. . . . . 6 ⊢ (card ‘A) = ∩{y ∈ On∣y ≈ A}
25	24	sseq2i 1525	. . . . 5 ⊢ (A ⊆ (card ‘A) ↔ A ⊆ ∩{y ∈ On∣y ≈ A})
26	23, 25	syl5bb 410	. . . 4 ⊢ (A ∈ On → (A ⊆ (card ‘A) ↔ ∀x ∈ On (A ≈ x → A ⊆ x)))
27	8, 26	bitrd 406	. . 3 ⊢ (A ∈ On → ((card ‘A) = A ↔ ∀x ∈ On (A ≈ x → A ⊆ x)))
28	27	pm5.32i 489	. 2 ⊢ ((A ∈ On ∧ (card ‘A) = A) ↔ (A ∈ On ∧ ∀x ∈ On (A ≈ x → A ⊆ x)))
29	4, 28	bitr 151	1 ⊢ ((card ‘A) = A ↔ (A ∈ On ∧ ∀x ∈ On (A ≈ x → A ⊆ x)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487  ∩cint 1965   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≈ cen 3271  cardccrd 3620

This theorem is referenced by:  ondomcard 3663

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

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