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Theorem psasym 17432
Description: A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
Assertion
Ref Expression
psasym ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)

Proof of Theorem psasym
StepHypRef Expression
1 pslem 17428 . . 3 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
21simp3d 1139 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵))
323impib 1109 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140   cuni 4589   class class class wbr 4805  PosetRelcps 17420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-res 5279  df-ps 17422
This theorem is referenced by:  psss  17436  ordtt1  21406  ordthauslem  21410
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