Theorem carddomi2 9923

Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10507, 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 9917	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
2	1	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3	2	biimpa 476	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4		0domg 9068	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∅ ≼ 𝐵)
5	4	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
6	3, 5	eqbrtrd 5129	. . 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 6871	. . . . 5 ⊢ (card‘𝐵) ∈ V
9		simprr 772	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10		ssdomg 8971	. . . . 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 9906	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
13	12	ad2antrr 726	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14		simprl 770	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15		ssn0 4367	. . . . . . 7 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
16	9, 14, 15	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17		ndmfv 6893	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
18	17	necon1ai 2952	. . . . . 6 ⊢ ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19		cardid2 9906	. . . . . 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 9083	. . . . . 6 ⊢ ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22		domen2 9084	. . . . . 6 ⊢ ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
23	21, 22	sylan9bb 509	. . . . 5 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
24	13, 20, 23	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
25	11, 24	mpbid 232	. . 3 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴 ≼ 𝐵)
26	25	expr 456	. 2 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
27	7, 26	pm2.61dane 3012	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447   ⊆ wss 3914  ∅c0 4296   class class class wbr 5107  dom cdm 5638  ‘cfv 6511   ≈ cen 8915   ≼ cdom 8916  cardccrd 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-er 8671  df-en 8919  df-dom 8920  df-card 9892

This theorem is referenced by:  carddom2  9930