Theorem disj4im 3276

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	⊢ ((𝐴 ∩ 𝐵) = ∅ → ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴)

Proof of Theorem disj4im

Step	Hyp	Ref	Expression
1		disj3 3272	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
2		eqcom 2042	. . 3 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ (𝐴 ∖ 𝐵) = 𝐴)
3	1, 2	bitri 173	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∖ 𝐵) = 𝐴)
4		dfpss2 3029	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ (𝐴 ∖ 𝐵) = 𝐴))
5	4	simprbi 260	. . 3 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 → ¬ (𝐴 ∖ 𝐵) = 𝐴)
6	5	con2i 557	. 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐴 → ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴)
7	3, 6	sylbi 114	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ∖ cdif 2914   ∩ cin 2916   ⊆ wss 2917   ⊊ wpss 2918  ∅c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-pss 2933  df-nul 3225

This theorem is referenced by: (None)