Theorem sspsstrir 3046

Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)

Assertion

Ref	Expression
sspsstrir	⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

Proof of Theorem sspsstrir

Step	Hyp	Ref	Expression
1		pssss 3039	. . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
2	1	orcd 652	. 2 ⊢ (𝐴 ⊊ 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
3		eqimss 2997	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
4	3	orcd 652	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
5		pssss 3039	. . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
6	5	olcd 653	. 2 ⊢ (𝐵 ⊊ 𝐴 → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
7	2, 4, 6	3jaoi 1198	1 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∨ wo 629   ∨ w3o 884   = wceq 1243   ⊆ 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-io 630  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-3or 886  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by: (None)