Theorem carden 3638

Description: Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size". This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank (see karden 3551).

Assertion

Ref	Expression
carden	⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) = (card ‘B) ↔ A ≈ B))

Proof of Theorem carden

Step	Hyp	Ref	Expression
1		breq2 2066	. . . . . 6 ⊢ ((card ‘A) = (card ‘B) → (A ≈ (card ‘A) ↔ A ≈ (card ‘B)))
2		cardid 3635	. . . . . . 7 ⊢ (card ‘B) ≈ B
3		entrt 3319	. . . . . . 7 ⊢ ((A ≈ (card ‘B) ∧ (card ‘B) ≈ B) → A ≈ B)
4	2, 3	mpan2 519	. . . . . 6 ⊢ (A ≈ (card ‘B) → A ≈ B)
5	1, 4	syl6bi 187	. . . . 5 ⊢ ((card ‘A) = (card ‘B) → (A ≈ (card ‘A) → A ≈ B))
6		cardid 3635	. . . . . 6 ⊢ (card ‘A) ≈ A
7		ensymg 3316	. . . . . 6 ⊢ (A ∈ C → ((card ‘A) ≈ A → A ≈ (card ‘A)))
8	6, 7	mpi 44	. . . . 5 ⊢ (A ∈ C → A ≈ (card ‘A))
9	5, 8	syl5 22	. . . 4 ⊢ ((card ‘A) = (card ‘B) → (A ∈ C → A ≈ B))
10	9	com12 13	. . 3 ⊢ (A ∈ C → ((card ‘A) = (card ‘B) → A ≈ B))
11	10	adantr 306	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) = (card ‘B) → A ≈ B))
12		ensymg 3316	. . . . . 6 ⊢ (B ∈ D → (A ≈ B → B ≈ A))
13		entrt 3319	. . . . . . . 8 ⊢ (((card ‘B) ≈ B ∧ B ≈ A) → (card ‘B) ≈ A)
14	2, 13	mpan 518	. . . . . . 7 ⊢ (B ≈ A → (card ‘B) ≈ A)
15		cardne 3637	. . . . . . . . 9 ⊢ ((card ‘B) ∈ (card ‘A) → ¬ (card ‘B) ≈ A)
16	15	con2i 89	. . . . . . . 8 ⊢ ((card ‘B) ≈ A → ¬ (card ‘B) ∈ (card ‘A))
17		cardon 3634	. . . . . . . . 9 ⊢ (card ‘A) ∈ On
18		cardon 3634	. . . . . . . . 9 ⊢ (card ‘B) ∈ On
19		ontri1 2232	. . . . . . . . 9 ⊢ (((card ‘A) ∈ On ∧ (card ‘B) ∈ On) → ((card ‘A) ⊆ (card ‘B) ↔ ¬ (card ‘B) ∈ (card ‘A)))
20	17, 18, 19	mp2an 520	. . . . . . . 8 ⊢ ((card ‘A) ⊆ (card ‘B) ↔ ¬ (card ‘B) ∈ (card ‘A))
21	16, 20	sylibr 175	. . . . . . 7 ⊢ ((card ‘B) ≈ A → (card ‘A) ⊆ (card ‘B))
22	14, 21	syl 12	. . . . . 6 ⊢ (B ≈ A → (card ‘A) ⊆ (card ‘B))
23	12, 22	syl6 23	. . . . 5 ⊢ (B ∈ D → (A ≈ B → (card ‘A) ⊆ (card ‘B)))
24		entrt 3319	. . . . . . . 8 ⊢ (((card ‘A) ≈ A ∧ A ≈ B) → (card ‘A) ≈ B)
25	6, 24	mpan 518	. . . . . . 7 ⊢ (A ≈ B → (card ‘A) ≈ B)
26		cardne 3637	. . . . . . . . 9 ⊢ ((card ‘A) ∈ (card ‘B) → ¬ (card ‘A) ≈ B)
27	26	con2i 89	. . . . . . . 8 ⊢ ((card ‘A) ≈ B → ¬ (card ‘A) ∈ (card ‘B))
28		ontri1 2232	. . . . . . . . 9 ⊢ (((card ‘B) ∈ On ∧ (card ‘A) ∈ On) → ((card ‘B) ⊆ (card ‘A) ↔ ¬ (card ‘A) ∈ (card ‘B)))
29	18, 17, 28	mp2an 520	. . . . . . . 8 ⊢ ((card ‘B) ⊆ (card ‘A) ↔ ¬ (card ‘A) ∈ (card ‘B))
30	27, 29	sylibr 175	. . . . . . 7 ⊢ ((card ‘A) ≈ B → (card ‘B) ⊆ (card ‘A))
31	25, 30	syl 12	. . . . . 6 ⊢ (A ≈ B → (card ‘B) ⊆ (card ‘A))
32	31	a1i 7	. . . . 5 ⊢ (B ∈ D → (A ≈ B → (card ‘B) ⊆ (card ‘A)))
33	23, 32	jcad 455	. . . 4 ⊢ (B ∈ D → (A ≈ B → ((card ‘A) ⊆ (card ‘B) ∧ (card ‘B) ⊆ (card ‘A))))
34		eqss 1516	. . . 4 ⊢ ((card ‘A) = (card ‘B) ↔ ((card ‘A) ⊆ (card ‘B) ∧ (card ‘B) ⊆ (card ‘A)))
35	33, 34	syl6ibr 186	. . 3 ⊢ (B ∈ D → (A ≈ B → (card ‘A) = (card ‘B)))
36	35	adantl 305	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (A ≈ B → (card ‘A) = (card ‘B)))
37	11, 36	impbid 397	1 ⊢ ((A ∈ C ∧ B ∈ D) → ((card ‘A) = (card ‘B) ↔ A ≈ B))

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

This theorem is referenced by:  cardeq0 3639  cardsn 3640  carddom 3642  cardsdom 3643  cardcard 3655  cfom 3710

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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