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Theorem sspsstri 3371
 Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
sspsstri ((A B B A) ↔ (AB A = B BA))

Proof of Theorem sspsstri
StepHypRef Expression
1 or32 513 . 2 (((AB BA) A = B) ↔ ((AB A = B) BA))
2 sspss 3368 . . . 4 (A B ↔ (AB A = B))
3 sspss 3368 . . . . 5 (B A ↔ (BA B = A))
4 eqcom 2355 . . . . . 6 (B = AA = B)
54orbi2i 505 . . . . 5 ((BA B = A) ↔ (BA A = B))
63, 5bitri 240 . . . 4 (B A ↔ (BA A = B))
72, 6orbi12i 507 . . 3 ((A B B A) ↔ ((AB A = B) (BA A = B)))
8 orordir 517 . . 3 (((AB BA) A = B) ↔ ((AB A = B) (BA A = B)))
97, 8bitr4i 243 . 2 ((A B B A) ↔ ((AB BA) A = B))
10 df-3or 935 . 2 ((AB A = B BA) ↔ ((AB A = B) BA))
111, 9, 103bitr4i 268 1 ((A B B A) ↔ (AB A = B BA))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∨ w3o 933   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by: (None)
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