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Theorem carddomi2 8986
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9568, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddomi2 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))

Proof of Theorem carddomi2
StepHypRef Expression
1 cardnueq0 8980 . . . . . 6 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
21adantr 472 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
32biimpa 502 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4 0domg 8252 . . . . 5 (𝐵𝑉 → ∅ ≼ 𝐵)
54ad2antlr 765 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
63, 5eqbrtrd 4826 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴𝐵)
76a1d 25 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
8 fvex 6362 . . . . 5 (card‘𝐵) ∈ V
9 simprr 813 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10 ssdomg 8167 . . . . 5 ((card‘𝐵) ∈ V → ((card‘𝐴) ⊆ (card‘𝐵) → (card‘𝐴) ≼ (card‘𝐵)))
118, 9, 10mpsyl 68 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≼ (card‘𝐵))
12 cardid2 8969 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
1312ad2antrr 764 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14 simprl 811 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15 ssn0 4119 . . . . . . 7 (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
169, 14, 15syl2anc 696 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17 ndmfv 6379 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
1817necon1ai 2959 . . . . . 6 ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19 cardid2 8969 . . . . . 6 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
2016, 18, 193syl 18 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
21 domen1 8267 . . . . . 6 ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22 domen2 8268 . . . . . 6 ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴𝐵))
2321, 22sylan9bb 738 . . . . 5 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2413, 20, 23syl2anc 696 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2511, 24mpbid 222 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴𝐵)
2625expr 644 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
277, 26pm2.61dane 3019 1 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  wss 3715  c0 4058   class class class wbr 4804  dom cdm 5266  cfv 6049  cen 8118  cdom 8119  cardccrd 8951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-er 7911  df-en 8122  df-dom 8123  df-card 8955
This theorem is referenced by:  carddom2  8993
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