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Theorem disj4im 3303
 Description: A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.)
Assertion
Ref Expression
disj4im ((𝐴𝐵) = ∅ → ¬ (𝐴𝐵) ⊊ 𝐴)

Proof of Theorem disj4im
StepHypRef Expression
1 disj3 3299 . . 3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eqcom 2058 . . 3 (𝐴 = (𝐴𝐵) ↔ (𝐴𝐵) = 𝐴)
31, 2bitri 177 . 2 ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) = 𝐴)
4 dfpss2 3056 . . . 4 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ ¬ (𝐴𝐵) = 𝐴))
54simprbi 264 . . 3 ((𝐴𝐵) ⊊ 𝐴 → ¬ (𝐴𝐵) = 𝐴)
65con2i 567 . 2 ((𝐴𝐵) = 𝐴 → ¬ (𝐴𝐵) ⊊ 𝐴)
73, 6sylbi 118 1 ((𝐴𝐵) = ∅ → ¬ (𝐴𝐵) ⊊ 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1259   ∖ cdif 2941   ∩ cin 2943   ⊆ wss 2944   ⊊ wpss 2945  ∅c0 3251 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2947  df-in 2951  df-pss 2960  df-nul 3252 This theorem is referenced by: (None)
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