Theorem sspssn 3048

Description: Like pssn2lp 3045 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)

Assertion

Ref	Expression
sspssn	⊢ ¬ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴)

Proof of Theorem sspssn

Step	Hyp	Ref	Expression
1		pm3.24 627	. 2 ⊢ ¬ (𝐵 ⊊ 𝐴 ∧ ¬ 𝐵 ⊊ 𝐴)
2		ssnpss 3047	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)
3	2	anim2i 324	. . 3 ⊢ ((𝐵 ⊊ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐵 ⊊ 𝐴 ∧ ¬ 𝐵 ⊊ 𝐴))
4	3	ancoms 255	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴) → (𝐵 ⊊ 𝐴 ∧ ¬ 𝐵 ⊊ 𝐴))
5	1, 4	mto 588	1 ⊢ ¬ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 97   ⊆ wss 2917   ⊊ wpss 2918

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by:  sspsstr  3050  psssstr  3051