Theorem soirri 3434

Description: A strict order relation is irreflexive.

Hypotheses

Ref	Expression
soi.1	⊢ A ∈ V
soi.2	⊢ R Or S
soi.3	⊢ R ⊆ (S × S)

Assertion

Ref	Expression
soirri	⊢ ¬ ARA

Proof of Theorem soirri

Step	Hyp	Ref	Expression
1		soi.2	. . . 4 ⊢ R Or S
2		sonr 2850	. . . 4 ⊢ ((R Or S ⋀ A ∈ S) → ¬ ARA)
3	1, 2	mpan 694	. . 3 ⊢ (A ∈ S → ¬ ARA)
4	3	adantl 388	. 2 ⊢ ((A ∈ S ⋀ A ∈ S) → ¬ ARA)
5		soi.1	. . . 4 ⊢ A ∈ V
6		soi.3	. . . 4 ⊢ R ⊆ (S × S)
7	5, 6	brel 3218	. . 3 ⊢ (ARA → (A ∈ S ⋀ A ∈ S))
8	7	con3i 98	. 2 ⊢ (¬ (A ∈ S ⋀ A ∈ S) → ¬ ARA)
9	4, 8	pm2.61i 126	1 ⊢ ¬ ARA

Syntax hints:  ¬ wn 2   ⋀ wa 223   ∈ wcel 956  Vcvv 1807   ⊆ wss 2043   class class class wbr 2614   Or wor 2834   × cxp 3163

This theorem is referenced by:  son2lpi 3436  ltrpq 5065  1pr 5097  ltapr 5131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-po 2835  df-so 2845  df-xp 3179

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