MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  carddomi2 Structured version   Visualization version   GIF version

Theorem carddomi2 10008
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10592, 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 10002 . . . . . 6 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
21adantr 480 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
32biimpa 476 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4 0domg 9139 . . . . 5 (𝐵𝑉 → ∅ ≼ 𝐵)
54ad2antlr 727 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
63, 5eqbrtrd 5170 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴𝐵)
76a1d 25 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
8 fvex 6920 . . . . 5 (card‘𝐵) ∈ V
9 simprr 773 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10 ssdomg 9039 . . . . 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 9991 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
1312ad2antrr 726 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14 simprl 771 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15 ssn0 4410 . . . . . . 7 (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
169, 14, 15syl2anc 584 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17 ndmfv 6942 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
1817necon1ai 2966 . . . . . 6 ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19 cardid2 9991 . . . . . 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 9158 . . . . . 6 ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22 domen2 9159 . . . . . 6 ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴𝐵))
2321, 22sylan9bb 509 . . . . 5 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2413, 20, 23syl2anc 584 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2511, 24mpbid 232 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴𝐵)
2625expr 456 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
277, 26pm2.61dane 3027 1 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148  dom cdm 5689  cfv 6563  cen 8981  cdom 8982  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-card 9977
This theorem is referenced by:  carddom2  10015
  Copyright terms: Public domain W3C validator