Theorem carddomi2 8796

Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9376, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
carddomi2	⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))

Proof of Theorem carddomi2

Step	Hyp	Ref	Expression
1		cardnueq0 8790	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
2	1	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3	2	biimpa 501	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4		0domg 8087	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∅ ≼ 𝐵)
5	4	ad2antlr 763	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
6	3, 5	eqbrtrd 4675	. . 3 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 ≼ 𝐵)
7	6	a1d 25	. 2 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
8		fvex 6201	. . . . 5 ⊢ (card‘𝐵) ∈ V
9		simprr 796	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10		ssdomg 8001	. . . . 5 ⊢ ((card‘𝐵) ∈ V → ((card‘𝐴) ⊆ (card‘𝐵) → (card‘𝐴) ≼ (card‘𝐵)))
11	8, 9, 10	mpsyl 68	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≼ (card‘𝐵))
12		cardid2 8779	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
13	12	ad2antrr 762	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14		simprl 794	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15		ssn0 3976	. . . . . . 7 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
16	9, 14, 15	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17		ndmfv 6218	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
18	17	necon1ai 2821	. . . . . 6 ⊢ ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19		cardid2 8779	. . . . . 6 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
20	16, 18, 19	3syl 18	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
21		domen1 8102	. . . . . 6 ⊢ ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22		domen2 8103	. . . . . 6 ⊢ ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
23	21, 22	sylan9bb 736	. . . . 5 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
24	13, 20, 23	syl2anc 693	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
25	11, 24	mpbid 222	. . 3 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴 ≼ 𝐵)
26	25	expr 643	. 2 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
27	7, 26	pm2.61dane 2881	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653  dom cdm 5114  ‘cfv 5888   ≈ cen 7952   ≼ cdom 7953  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-card 8765

This theorem is referenced by:  carddom2  8803