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Theorem carddomi2 8679
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9255, 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 8673 . . . . . 6 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
21adantr 480 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
32biimpa 500 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4 0domg 7972 . . . . 5 (𝐵𝑉 → ∅ ≼ 𝐵)
54ad2antlr 759 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
63, 5eqbrtrd 4605 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴𝐵)
76a1d 25 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
8 fvex 6113 . . . . 5 (card‘𝐵) ∈ V
9 simprr 792 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10 ssdomg 7887 . . . . 5 ((card‘𝐵) ∈ V → ((card‘𝐴) ⊆ (card‘𝐵) → (card‘𝐴) ≼ (card‘𝐵)))
118, 9, 10mpsyl 66 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≼ (card‘𝐵))
12 cardid2 8662 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
1312ad2antrr 758 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14 simprl 790 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15 ssn0 3928 . . . . . . 7 (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
169, 14, 15syl2anc 691 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17 ndmfv 6128 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
1817necon1ai 2809 . . . . . 6 ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19 cardid2 8662 . . . . . 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 7987 . . . . . 6 ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22 domen2 7988 . . . . . 6 ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴𝐵))
2321, 22sylan9bb 732 . . . . 5 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2413, 20, 23syl2anc 691 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2511, 24mpbid 221 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴𝐵)
2625expr 641 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
277, 26pm2.61dane 2869 1 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  wss 3540  c0 3874   class class class wbr 4583  dom cdm 5038  cfv 5804  cen 7838  cdom 7839  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-card 8648
This theorem is referenced by:  carddom2  8686
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