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Theorem carddomi2 9911
Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10495, 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 9905 . . . . . 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 9047 . . . . 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 5128 . . 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 6856 . . . . 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 8943 . . . . 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 9894 . . . . . 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 4361 . . . . . . 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 6878 . . . . . . 7 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
1817necon1ai 2968 . . . . . 6 ((cardβ€˜π΅) β‰  βˆ… β†’ 𝐡 ∈ dom card)
19 cardid2 9894 . . . . . 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 9066 . . . . . 6 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ ((cardβ€˜π΄) β‰Ό (cardβ€˜π΅) ↔ 𝐴 β‰Ό (cardβ€˜π΅)))
22 domen2 9067 . . . . . 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 3029 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 2940  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283   class class class wbr 5106  dom cdm 5634  β€˜cfv 6497   β‰ˆ cen 8883   β‰Ό cdom 8884  cardccrd 9876
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-dom 8888  df-card 9880
This theorem is referenced by:  carddom2  9918
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