Theorem nssr 2997

Description: Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)

Assertion

Ref	Expression
nssr	⊢ (∃x(x ∈ A ∧ ¬ x ∈ B) → ¬ A ⊆ B)

Distinct variable groups:   x,A   x,B

Proof of Theorem nssr

Step	Hyp	Ref	Expression
1		exanaliim 1535	. 2 ⊢ (∃x(x ∈ A ∧ ¬ x ∈ B) → ¬ ∀x(x ∈ A → x ∈ B))
2		dfss2 2928	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
3	1, 2	sylnibr 601	1 ⊢ (∃x(x ∈ A ∧ ¬ x ∈ B) → ¬ A ⊆ B)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911

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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by: (None)