Theorem carddom2 9999

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10576, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
carddom2	⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))

Proof of Theorem carddom2

Step	Hyp	Ref	Expression
1		carddomi2 9992	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
2		brdom2 9004	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))
3		cardon 9966	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
4	3	onelssi 6479	. . . . . . 7 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5		carddomi2 9992	. . . . . . . 8 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴))
6	5	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴))
7		domnsym 9121	. . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵)
8	4, 6, 7	syl56 36	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴 ≺ 𝐵))
9	8	con2d 134	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10		cardon 9966	. . . . . 6 ⊢ (card‘𝐵) ∈ On
11		ontri1 6397	. . . . . 6 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
12	3, 10, 11	mp2an 692	. . . . 5 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
13	9, 12	imbitrrdi 252	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14		carden2b 9989	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
15		eqimss 4022	. . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (card‘𝐴) ⊆ (card‘𝐵))
16	14, 15	syl 17	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵))
17	16	a1i 11	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≈ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
18	13, 17	jaod 859	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
19	2, 18	biimtrid 242	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
20	1, 19	impbid 212	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931   class class class wbr 5123  dom cdm 5665  Oncon0 6363  ‘cfv 6541   ≈ cen 8964   ≼ cdom 8965   ≺ csdm 8966  cardccrd 9957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-card 9961

This theorem is referenced by:  carduni  10003  carden2  10009  cardsdom2  10010  domtri2  10011  infxpidm2  10039  cardaleph  10111  infenaleph  10113  alephinit  10117  ficardun2  10224  ackbij2  10264  cfflb  10281  fin1a2lem9  10430  carddom  10576  pwfseqlem5  10685  hashdom  14400  minregex2  43510