Theorem frirr 2176

Description: A founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
frirr	|- ((R Fr A /\ x e. A) -> -. xRx)

Distinct variable group(s):   x,R

Proof of Theorem frirr

Step	Hyp	Ref	Expression
1		visset 1350	. . . . 5 |- x e. V
2	1	snnz 1846	. . . 4 |- -. {x} = (/)
3		snex 1859	. . . . . . 7 |- {x} e. V
4	3	frc 2172	. . . . . 6 |- (R Fr A -> (({x} (_ A /\ -. {x} = (/)) -> E.y e. {x} ({x} i^i {z | zRy}) = (/)))
5	4	exp3a 292	. . . . 5 |- (R Fr A -> ({x} (_ A -> (-. {x} = (/) -> E.y e. {x} ({x} i^i {z | zRy}) = (/))))
6	1	snss 1849	. . . . 5 |- (x e. A <-> {x} (_ A)
7	5, 6	syl5ib 181	. . . 4 |- (R Fr A -> (x e. A -> (-. {x} = (/) -> E.y e. {x} ({x} i^i {z | zRy}) = (/))))
8	2, 7	mpii 45	. . 3 |- (R Fr A -> (x e. A -> E.y e. {x} ({x} i^i {z | zRy}) = (/)))
9		elsn 1820	. . . . 5 |- (y e. {x} <-> y = x)
10		breq2 2066	. . . . . . . . 9 |- (y = x -> (zRy <-> zRx))
11	10	biabdv 1183	. . . . . . . 8 |- (y = x -> {z | zRy} = {z | zRx})
12	11	ineq2d 1645	. . . . . . 7 |- (y = x -> ({x} i^i {z | zRy}) = ({x} i^i {z | zRx}))
13	12	cleq1d 1109	. . . . . 6 |- (y = x -> (({x} i^i {z | zRy}) = (/) <-> ({x} i^i {z | zRx}) = (/)))
14		breq1 2065	. . . . . . . . . . . 12 |- (z = x -> (zRx <-> xRx))
15	1, 14	elab 1415	. . . . . . . . . . 11 |- (x e. {z | zRx} <-> xRx)
16	15	biimpr 134	. . . . . . . . . 10 |- (xRx -> x e. {z | zRx})
17	1	snid 1830	. . . . . . . . . 10 |- x e. {x}
18	16, 17	jctil 240	. . . . . . . . 9 |- (xRx -> (x e. {x} /\ x e. {z | zRx}))
19		elin 1635	. . . . . . . . 9 |- (x e. ({x} i^i {z | zRx}) <-> (x e. {x} /\ x e. {z | zRx}))
20	18, 19	sylibr 175	. . . . . . . 8 |- (xRx -> x e. ({x} i^i {z | zRx}))
21		n0i 1712	. . . . . . . 8 |- (x e. ({x} i^i {z | zRx}) -> -. ({x} i^i {z | zRx}) = (/))
22	20, 21	syl 12	. . . . . . 7 |- (xRx -> -. ({x} i^i {z | zRx}) = (/))
23	22	con2i 89	. . . . . 6 |- (({x} i^i {z | zRx}) = (/) -> -. xRx)
24	13, 23	syl6bi 187	. . . . 5 |- (y = x -> (({x} i^i {z | zRy}) = (/) -> -. xRx))
25	9, 24	sylbi 174	. . . 4 |- (y e. {x} -> (({x} i^i {z | zRy}) = (/) -> -. xRx))
26	25	r19.23aiv 1284	. . 3 |- (E.y e. {x} ({x} i^i {z | zRy}) = (/) -> -. xRx)
27	8, 26	syl6 23	. 2 |- (R Fr A -> (x e. A -> -. xRx))
28	27	imp 277	1 |- ((R Fr A /\ x e. A) -> -. xRx)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707  {csn 1808   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  efrirr 2180  dfwe2 2187

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-fr 2169

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