Theorem psasym 8651

Description: A poset is antisymmetric.

Assertion

Ref	Expression
psasym	|- ((R e. Poset /\ ARB /\ BRA) -> A = B)

Proof of Theorem psasym

Step	Hyp	Ref	Expression
1		pslem 8647	. . . 4 |- (R e. Poset -> ((A e. V /\ B e. V /\ B e. V) -> (((ARB /\ BRB) -> ARB) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
2		simprr 415	. . . 4 |- ((((ARB /\ BRB) -> ARB) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B))) -> ((ARB /\ BRA) -> A = B))
3	1, 2	syl6 22	. . 3 |- (R e. Poset -> ((A e. V /\ B e. V /\ B e. V) -> ((ARB /\ BRA) -> A = B)))
4	3	3imp 827	. 2 |- ((R e. Poset /\ (A e. V /\ B e. V /\ B e. V) /\ (ARB /\ BRA)) -> A = B)
5		3simp1 788	. 2 |- ((R e. Poset /\ ARB /\ BRA) -> R e. Poset)
6		brrelex 3207	. . . . 5 |- ((Rel R /\ ARB) -> A e. V)
7	6	3adant3 799	. . . 4 |- ((Rel R /\ ARB /\ BRA) -> A e. V)
8		brrelex 3207	. . . . 5 |- ((Rel R /\ BRA) -> B e. V)
9	8	3adant2 798	. . . 4 |- ((Rel R /\ ARB /\ BRA) -> B e. V)
10	7, 9, 9	3jca 819	. . 3 |- ((Rel R /\ ARB /\ BRA) -> (A e. V /\ B e. V /\ B e. V))
11		psrel 8646	. . 3 |- (R e. Poset -> Rel R)
12	10, 11	syl3an1 859	. 2 |- ((R e. Poset /\ ARB /\ BRA) -> (A e. V /\ B e. V /\ B e. V))
13		3simpc 787	. 2 |- ((R e. Poset /\ ARB /\ BRA) -> (ARB /\ BRA))
14	4, 5, 12, 13	syl3anc 858	1 |- ((R e. Poset /\ ARB /\ BRA) -> A = B)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  Vcvv 1811  U.cuni 2503   class class class wbr 2619  Rel wrel 3175  Posetcps 8633

This theorem is referenced by:  spwmo 8656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ps 8639

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