Theorem nssinpss 4159

Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
nssinpss	⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 4131	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	biantrur 531	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
3		df-ss 3880	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
4	3	necon3bbii 3033	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ≠ 𝐴)
5		df-pss 3882	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
6	2, 4, 5	3bitr4i 304	1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ≠ wne 2986   ∩ cin 3864   ⊆ wss 3865   ⊊ wpss 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-v 3442  df-in 3872  df-ss 3880  df-pss 3882

This theorem is referenced by:  fbfinnfr  22137  chrelat2i  29829