Theorem frirr 2919

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)

Proof of Theorem frirr

Step	Hyp	Ref	Expression
1		visset 1809	. . . . 5 |- x e. V
2	1	snnz 2454	. . . 4 |- {x} =/= (/)
3		snex 2745	. . . . . . 7 |- {x} e. V
4	3	frc 2915	. . . . . 6 |- ((R Fr A /\ {x} (_ A /\ {x} =/= (/)) -> E.y e. {x} ({x} i^i {z | zRy}) = (/))
5	4	3exp 831	. . . . 5 |- (R Fr A -> ({x} (_ A -> ({x} =/= (/) -> E.y e. {x} ({x} i^i {z | zRy}) = (/))))
6	1	snss 2457	. . . . 5 |- (x e. A <-> {x} (_ A)
7	5, 6	syl5ib 206	. . . 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 2417	. . . . 5 |- (y e. {x} <-> y = x)
10		breq2 2618	. . . . . . . . 9 |- (y = x -> (zRy <-> zRx))
11	10	abbidv 1574	. . . . . . . 8 |- (y = x -> {z | zRy} = {z | zRx})
12	11	ineq2d 2213	. . . . . . 7 |- (y = x -> ({x} i^i {z | zRy}) = ({x} i^i {z | zRx}))
13	12	eqeq1d 1480	. . . . . 6 |- (y = x -> (({x} i^i {z | zRy}) = (/) <-> ({x} i^i {z | zRx}) = (/)))
14		breq1 2617	. . . . . . . . . . . 12 |- (z = x -> (zRx <-> xRx))
15	1, 14	elab 1893	. . . . . . . . . . 11 |- (x e. {z | zRx} <-> xRx)
16	15	biimpr 152	. . . . . . . . . 10 |- (xRx -> x e. {z | zRx})
17	1	snid 2431	. . . . . . . . . 10 |- x e. {x}
18	16, 17	jctil 292	. . . . . . . . 9 |- (xRx -> (x e. {x} /\ x e. {z | zRx}))
19		elin 2203	. . . . . . . . 9 |- (x e. ({x} i^i {z | zRx}) <-> (x e. {x} /\ x e. {z | zRx}))
20	18, 19	sylibr 200	. . . . . . . 8 |- (xRx -> x e. ({x} i^i {z | zRx}))
21		n0i 2281	. . . . . . . 8 |- (x e. ({x} i^i {z | zRx}) -> -. ({x} i^i {z | zRx}) = (/))
22	20, 21	syl 10	. . . . . . 7 |- (xRx -> -. ({x} i^i {z | zRx}) = (/))
23	22	con2i 97	. . . . . 6 |- (({x} i^i {z | zRx}) = (/) -> -. xRx)
24	13, 23	syl6bi 214	. . . . 5 |- (y = x -> (({x} i^i {z | zRy}) = (/) -> -. xRx))
25	9, 24	sylbi 199	. . . 4 |- (y e. {x} -> (({x} i^i {z | zRy}) = (/) -> -. xRx))
26	25	r19.23aiv 1740	. . 3 |- (E.y e. {x} ({x} i^i {z | zRy}) = (/) -> -. xRx)
27	8, 26	syl6 22	. 2 |- (R Fr A -> (x e. A -> -. xRx))
28	27	imp 350	1 |- ((R Fr A /\ x e. A) -> -. xRx)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461   =/= wne 1582  E.wrex 1643   i^i cin 2042   (_ wss 2043  (/)c0 2276  {csn 2405   class class class wbr 2614   Fr wfr 2910

This theorem is referenced by:  efrirr 2923  dfwe2 2930

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

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-rex 1647  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-fr 2912

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