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Theorem carddomi2 9390
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9968, 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 9384 . . . . . 6 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
21adantr 484 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
32biimpa 480 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4 0domg 8635 . . . . 5 (𝐵𝑉 → ∅ ≼ 𝐵)
54ad2antlr 726 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
63, 5eqbrtrd 5074 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴𝐵)
76a1d 25 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
8 fvex 6671 . . . . 5 (card‘𝐵) ∈ V
9 simprr 772 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10 ssdomg 8545 . . . . 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 9373 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
1312ad2antrr 725 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14 simprl 770 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15 ssn0 4336 . . . . . . 7 (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
169, 14, 15syl2anc 587 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17 ndmfv 6688 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
1817necon1ai 3041 . . . . . 6 ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19 cardid2 9373 . . . . . 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 8650 . . . . . 6 ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22 domen2 8651 . . . . . 6 ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴𝐵))
2321, 22sylan9bb 513 . . . . 5 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2413, 20, 23syl2anc 587 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2511, 24mpbid 235 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴𝐵)
2625expr 460 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
277, 26pm2.61dane 3101 1 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  Vcvv 3480  wss 3919  c0 4275   class class class wbr 5052  dom cdm 5542  cfv 6343  cen 8496  cdom 8497  cardccrd 9355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-er 8279  df-en 8500  df-dom 8501  df-card 9359
This theorem is referenced by:  carddom2  9397
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