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Theorem carddomi2 9967
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10551, 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 9961 . . . . . 6 (𝐴 ∈ dom card β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
21adantr 479 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
32biimpa 475 . . . 4 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) = βˆ…) β†’ 𝐴 = βˆ…)
4 0domg 9102 . . . . 5 (𝐡 ∈ 𝑉 β†’ βˆ… β‰Ό 𝐡)
54ad2antlr 723 . . . 4 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) = βˆ…) β†’ βˆ… β‰Ό 𝐡)
63, 5eqbrtrd 5169 . . 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 6903 . . . . 5 (cardβ€˜π΅) ∈ V
9 simprr 769 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))
10 ssdomg 8998 . . . . 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 9950 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
1312ad2antrr 722 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
14 simprl 767 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΄) β‰  βˆ…)
15 ssn0 4399 . . . . . . 7 (((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΄) β‰  βˆ…) β†’ (cardβ€˜π΅) β‰  βˆ…)
169, 14, 15syl2anc 582 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ ((cardβ€˜π΄) β‰  βˆ… ∧ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))) β†’ (cardβ€˜π΅) β‰  βˆ…)
17 ndmfv 6925 . . . . . . 7 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
1817necon1ai 2966 . . . . . 6 ((cardβ€˜π΅) β‰  βˆ… β†’ 𝐡 ∈ dom card)
19 cardid2 9950 . . . . . 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 9121 . . . . . 6 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ ((cardβ€˜π΄) β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό (cardβ€˜π΅)))
22 domen2 9122 . . . . . 6 ((cardβ€˜π΅) β‰ˆ 𝐡 β†’ (𝐴 β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2321, 22sylan9bb 508 . . . . 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 582 . . . 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 455 . 2 (((𝐴 ∈ dom card ∧ 𝐡 ∈ 𝑉) ∧ (cardβ€˜π΄) β‰  βˆ…) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) β†’ 𝐴 β‰Ό 𝐡))
277, 26pm2.61dane 3027 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 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  dom cdm 5675  β€˜cfv 6542   β‰ˆ cen 8938   β‰Ό cdom 8939  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-er 8705  df-en 8942  df-dom 8943  df-card 9936
This theorem is referenced by:  carddom2  9974
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