Theorem eldifbd 2924

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
eldifbd.1	⊢ (φ → A ∈ (B ∖ 𝐶))

Assertion

Ref	Expression
eldifbd	⊢ (φ → ¬ A ∈ 𝐶)

Proof of Theorem eldifbd

Step	Hyp	Ref	Expression
1		eldifbd.1	. . 3 ⊢ (φ → A ∈ (B ∖ 𝐶))
2		eldif 2921	. . 3 ⊢ (A ∈ (B ∖ 𝐶) ↔ (A ∈ B ∧ ¬ A ∈ 𝐶))
3	1, 2	sylib 127	. 2 ⊢ (φ → (A ∈ B ∧ ¬ A ∈ 𝐶))
4	3	simprd 107	1 ⊢ (φ → ¬ A ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1390   ∖ cdif 2908

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914

This theorem is referenced by: (None)