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Theorem psasym 17204
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 17200 . . 3 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
21simp3d 1074 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵))
323impib 1261 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989   cuni 4434   class class class wbr 4651  PosetRelcps 17192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-res 5124  df-ps 17194
This theorem is referenced by:  psss  17208  ordtt1  21177  ordthauslem  21181
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