Theorem carddomi2 9885

Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10467, 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 9879	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
2	1	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3	2	biimpa 477	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4		0domg 9032	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∅ ≼ 𝐵)
5	4	ad2antlr 733	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
6	3, 5	eqbrtrd 5094	. . 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 6840	. . . . 5 ⊢ (card‘𝐵) ∈ V
9		simprr 778	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10		ssdomg 8937	. . . . 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 9868	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
13	12	ad2antrr 732	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14		simprl 776	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15		ssn0 4332	. . . . . . 7 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
16	9, 14, 15	syl2anc 590	. . . . . 6 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17		ndmfv 6859	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
18	17	necon1ai 2961	. . . . . 6 ⊢ ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19		cardid2 9868	. . . . . 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 9047	. . . . . 6 ⊢ ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22		domen2 9048	. . . . . 6 ⊢ ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
23	21, 22	sylan9bb 514	. . . . 5 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
24	13, 20, 23	syl2anc 590	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
25	11, 24	mpbid 233	. . 3 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴 ≼ 𝐵)
26	25	expr 457	. 2 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
27	7, 26	pm2.61dane 3021	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431   ⊆ wss 3883  ∅c0 4261   class class class wbr 5072  dom cdm 5618  ‘cfv 6485   ≈ cen 8880   ≼ cdom 8881  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-er 8633  df-en 8884  df-dom 8885  df-card 9854

This theorem is referenced by:  carddom2  9892