Theorem papirr 7601

Description: An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)

Assertion

Ref	Expression
papirr	|- ( ( R Ap A /\ X e. A ) -> -. X R X )

Proof of Theorem papirr

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( x = X -> x = X )
2	1, 1	breq12d 4138	. . 3 |- ( x = X -> ( x R x <-> X R X ) )
3	2	notbid 677	. 2 |- ( x = X -> ( -. x R x <-> -. X R X ) )
4		df-pap 7598	. . . . 5 |- ( R Ap A <-> ( ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) /\ ( A. x e. A A. y e. A ( x R y -> y R x ) /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) ) ) )
5	4	simplbi 274	. . . 4 |- ( R Ap A -> ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) )
6	5	simprd 114	. . 3 |- ( R Ap A -> A. x e. A -. x R x )
7	6	adantr 276	. 2 |- ( ( R Ap A /\ X e. A ) -> A. x e. A -. x R x )
8		simpr 110	. 2 |- ( ( R Ap A /\ X e. A ) -> X e. A )
9	3, 7, 8	rspcdva 2934	1 |- ( ( R Ap A /\ X e. A ) -> -. X R X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   class class class wbr 4125   X. cxp 4767   Ap wap 7597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-pap 7598

This theorem is referenced by:  aprnzr  14572