Theorem carddomi2 9978

Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10565, 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 9972	. . . . . 6 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
2	1	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3	2	biimpa 482	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4		0domg 9105	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∅ ≼ 𝐵)
5	4	ad2antlr 740	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
6	3, 5	eqbrtrd 5127	. . 3 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 ≼ 𝐵)
7	6	a1d 26	. 2 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
8		fvex 6892	. . . . 5 ⊢ (card‘𝐵) ∈ V
9		simprr 785	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10		ssdomg 9009	. . . . 5 ⊢ ((card‘𝐵) ∈ V → ((card‘𝐴) ⊆ (card‘𝐵) → (card‘𝐴) ≼ (card‘𝐵)))
11	8, 9, 10	mpsyl 69	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≼ (card‘𝐵))
12		cardid2 9961	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
13	12	ad2antrr 739	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14		simprl 783	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15		ssn0 4355	. . . . . . 7 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
16	9, 14, 15	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17		ndmfv 6911	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
18	17	necon1ai 2982	. . . . . 6 ⊢ ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19		cardid2 9961	. . . . . 6 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
20	16, 18, 19	3syl 19	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
21		domen1 9120	. . . . . 6 ⊢ ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22		domen2 9121	. . . . . 6 ⊢ ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
23	21, 22	sylan9bb 519	. . . . 5 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
24	13, 20, 23	syl2anc 596	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
25	11, 24	mpbid 235	. . 3 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴 ≼ 𝐵)
26	25	expr 462	. 2 ⊢ (((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
27	7, 26	pm2.61dane 3042	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  dom cdm 5655  ‘cfv 6533   ≈ cen 8952   ≼ cdom 8953  cardccrd 9943

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-er 8699  df-en 8956  df-dom 8957  df-card 9947

This theorem is used by:  carddom2  9985