Theorem carddom 3642

Description: Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation (i) of [Quine] p. 232.

Assertion

Ref	Expression
carddom	⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) ⊆ (card ‘B) ↔ A ≼ B))

Proof of Theorem carddom

Step	Hyp	Ref	Expression
1		carddomi 3641	. . 3 ⊢ (A ∈ C → ((card ‘A) ⊆ (card ‘B) → A ≼ B))
2	1	adantr 306	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) ⊆ (card ‘B) → A ≼ B))
3		carddomi 3641	. . . . . . . . 9 ⊢ (B ∈ D → ((card ‘B) ⊆ (card ‘A) → B ≼ A))
4		cardon 3634	. . . . . . . . . 10 ⊢ (card ‘A) ∈ On
5	4	onelss 2348	. . . . . . . . 9 ⊢ ((card ‘B) ∈ (card ‘A) → (card ‘B) ⊆ (card ‘A))
6	3, 5	syl5 22	. . . . . . . 8 ⊢ (B ∈ D → ((card ‘B) ∈ (card ‘A) → B ≼ A))
7		domnsym 3365	. . . . . . . 8 ⊢ (B ≼ A → ¬ A ≺ B)
8	6, 7	syl6 23	. . . . . . 7 ⊢ (B ∈ D → ((card ‘B) ∈ (card ‘A) → ¬ A ≺ B))
9	8	con2d 83	. . . . . 6 ⊢ (B ∈ D → (A ≺ B → ¬ (card ‘B) ∈ (card ‘A)))
10		cardon 3634	. . . . . . 7 ⊢ (card ‘B) ∈ On
11		ontri1 2232	. . . . . . 7 ⊢ (((card ‘A) ∈ On ∧ (card ‘B) ∈ On) → ((card ‘A) ⊆ (card ‘B) ↔ ¬ (card ‘B) ∈ (card ‘A)))
12	4, 10, 11	mp2an 520	. . . . . 6 ⊢ ((card ‘A) ⊆ (card ‘B) ↔ ¬ (card ‘B) ∈ (card ‘A))
13	9, 12	syl6ibr 186	. . . . 5 ⊢ (B ∈ D → (A ≺ B → (card ‘A) ⊆ (card ‘B)))
14	13	adantl 305	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → (A ≺ B → (card ‘A) ⊆ (card ‘B)))
15		carden 3638	. . . . 5 ⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) = (card ‘B) ↔ A ≈ B))
16		eqimss 1548	. . . . 5 ⊢ ((card ‘A) = (card ‘B) → (card ‘A) ⊆ (card ‘B))
17	15, 16	syl6bir 188	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → (A ≈ B → (card ‘A) ⊆ (card ‘B)))
18	14, 17	jaod 329	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → ((A ≺ B ∨ A ≈ B) → (card ‘A) ⊆ (card ‘B)))
19		brdom2 3292	. . 3 ⊢ (A ≼ B ↔ (A ≺ B ∨ A ≈ B))
20	18, 19	syl5ib 181	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (A ≼ B → (card ‘A) ⊆ (card ‘B)))
21	2, 20	impbid 397	1 ⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) ⊆ (card ‘B) ↔ A ≼ B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≈ cen 3271   ≼ cdom 3272   ≺ csdm 3273  cardccrd 3620

This theorem is referenced by:  cardsdom 3643  domtri 3644  carduni 3664  cardprc 3667  cardaleph 3690

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-dom 3275  df-sdom 3276  df-card 3623

metamath.org