Theorem carddom2 8803

Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9376, 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 8796	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
2		brdom2 7985	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))
3		cardon 8770	. . . . . . . 8 ⊢ (card‘𝐴) ∈ On
4	3	onelssi 5836	. . . . . . 7 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5		carddomi2 8796	. . . . . . . 8 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴))
6	5	ancoms 469	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴))
7		domnsym 8086	. . . . . . 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 129	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10		cardon 8770	. . . . . 6 ⊢ (card‘𝐵) ∈ On
11		ontri1 5757	. . . . . 6 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
12	3, 10, 11	mp2an 708	. . . . 5 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
13	9, 12	syl6ibr 242	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14		carden2b 8793	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
15		eqimss 3657	. . . . . 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 395	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
19	2, 18	syl5bi 232	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
20	1, 19	impbid 202	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  Oncon0 5723  ‘cfv 5888   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  carduni  8807  carden2  8813  cardsdom2  8814  domtri2  8815  infxpidm2  8840  cardaleph  8912  infenaleph  8914  alephinit  8918  ficardun2  9025  ackbij2  9065  cfflb  9081  fin1a2lem9  9230  carddom  9376  pwfseqlem5  9485  hashdom  13168