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Theorem carddomi2 9965
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10549, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddomi2 ((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡))

Proof of Theorem carddomi2
StepHypRef Expression
1 cardnueq0 9959 . . . . . 6 (𝐴 ∈ dom card β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
21adantr 482 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
32biimpa 478 . . . 4 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) = βˆ…) β†’ 𝐴 = βˆ…)
4 0domg 9100 . . . . 5 (𝐡 ∈ 𝑉 β†’ βˆ… β‰Ό 𝐡)
54ad2antlr 726 . . . 4 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) = βˆ…) β†’ βˆ… β‰Ό 𝐡)
63, 5eqbrtrd 5171 . . 3 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) = βˆ…) β†’ 𝐴 β‰Ό 𝐡)
76a1d 25 . 2 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) = βˆ…) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡))
8 fvex 6905 . . . . 5 (cardβ€˜π΅) ∈ V
9 simprr 772 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))
10 ssdomg 8996 . . . . 5 ((cardβ€˜π΅) ∈ V β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ (cardβ€˜π΄) β‰Ό (cardβ€˜π΅)))
118, 9, 10mpsyl 68 . . . 4 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) β‰Ό (cardβ€˜π΅))
12 cardid2 9948 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
1312ad2antrr 725 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
14 simprl 770 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) β‰  βˆ…)
15 ssn0 4401 . . . . . . 7 (((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΄) β‰  βˆ…) β†’ (cardβ€˜π΅) β‰  βˆ…)
169, 14, 15syl2anc 585 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΅) β‰  βˆ…)
17 ndmfv 6927 . . . . . . 7 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
1817necon1ai 2969 . . . . . 6 ((cardβ€˜π΅) β‰  βˆ… β†’ 𝐡 ∈ dom card)
19 cardid2 9948 . . . . . 6 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
2016, 18, 193syl 18 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
21 domen1 9119 . . . . . 6 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ ((cardβ€˜π΄) β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό (cardβ€˜π΅)))
22 domen2 9120 . . . . . 6 ((cardβ€˜π΅) β‰ˆ 𝐡 β†’ (𝐴 β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2321, 22sylan9bb 511 . . . . 5 (((cardβ€˜π΄) β‰ˆ 𝐴 ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ ((cardβ€˜π΄) β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2413, 20, 23syl2anc 585 . . . 4 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ ((cardβ€˜π΄) β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2511, 24mpbid 231 . . 3 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ 𝐴 β‰Ό 𝐡)
2625expr 458 . 2 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) β‰  βˆ…) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡))
277, 26pm2.61dane 3030 1 ((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544   β‰ˆ cen 8936   β‰Ό cdom 8937  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-card 9934
This theorem is referenced by:  carddom2  9972
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