Theorem carddom2 10058

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10638, 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 10051	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
2		brdom2 9009	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))
3		cardon 10025	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
4	3	onelssi 6479	. . . . . . 7 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5		carddomi2 10051	. . . . . . . 8 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴))
6	5	ancoms 464	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴))
7		domnsym 9122	. . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵)
8	4, 6, 7	syl56 37	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴 ≺ 𝐵))
9	8	con2d 135	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10		cardon 10025	. . . . . 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 705	. . . . 5 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
13	9, 12	imbitrrdi 255	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14		carden2b 10048	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
15		eqimss 3966	. . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (card‘𝐴) ⊆ (card‘𝐵))
16	14, 15	syl 18	. . . . 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 873	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
19	2, 18	biimtrid 245	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
20	1, 19	impbid 215	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  dom cdm 5650  Oncon0 6362  ‘cfv 6538   ≈ cen 8970   ≼ cdom 8971   ≺ csdm 8972  cardccrd 10016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-card 10020

This theorem is used by:  carduni  10062  carden2  10068  cardsdom2  10069  domtri2  10070  infxpidm2  10096  cardaleph  10168  infenaleph  10170  alephinit  10174  ficardun2  10280  ackbij2  10320  cfflb  10337  fin1a2lem9  10486  carddom  10638  pwfseqlem5  10748  hashdom  14523  minregex2  44536