Theorem frirr 2979

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 1851	. . . . 5 |- x e. V
2	1	snnz 2506	. . . 4 |- {x} =/= (/)
3		snex 2802	. . . . . . 7 |- {x} e. V
4	3	frc 2975	. . . . . 6 |- ((R Fr A /\ {x} (_ A /\ {x} =/= (/)) -> E.y e. {x} ({x} i^i {z | zRy}) = (/))
5	4	3exp 835	. . . . 5 |- (R Fr A -> ({x} (_ A -> ({x} =/= (/) -> E.y e. {x} ({x} i^i {z | zRy}) = (/))))
6	1	snss 2509	. . . . 5 |- (x e. A <-> {x} (_ A)
7	5, 6	syl5ib 204	. . . 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 2466	. . . . 5 |- (y e. {x} <-> y = x)
10		breq2 2673	. . . . . . . . 9 |- (y = x -> (zRy <-> zRx))
11	10	abbidv 1614	. . . . . . . 8 |- (y = x -> {z | zRy} = {z | zRx})
12	11	ineq2d 2261	. . . . . . 7 |- (y = x -> ({x} i^i {z | zRy}) = ({x} i^i {z | zRx}))
13	12	eqeq1d 1520	. . . . . 6 |- (y = x -> (({x} i^i {z | zRy}) = (/) <-> ({x} i^i {z | zRx}) = (/)))
14		breq1 2672	. . . . . . . . . . . 12 |- (z = x -> (zRx <-> xRx))
15	1, 14	elab 1935	. . . . . . . . . . 11 |- (x e. {z | zRx} <-> xRx)
16	15	biimpri 150	. . . . . . . . . 10 |- (xRx -> x e. {z | zRx})
17	1	snid 2480	. . . . . . . . . 10 |- x e. {x}
18	16, 17	jctil 290	. . . . . . . . 9 |- (xRx -> (x e. {x} /\ x e. {z | zRx}))
19		elin 2251	. . . . . . . . 9 |- (x e. ({x} i^i {z | zRx}) <-> (x e. {x} /\ x e. {z | zRx}))
20	18, 19	sylibr 198	. . . . . . . 8 |- (xRx -> x e. ({x} i^i {z | zRx}))
21		n0i 2329	. . . . . . . 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 212	. . . . 5 |- (y = x -> (({x} i^i {z | zRy}) = (/) -> -. xRx))
25	9, 24	sylbi 197	. . . 4 |- (y e. {x} -> (({x} i^i {z | zRy}) = (/) -> -. xRx))
26	25	r19.23aiv 1781	. . 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 348	1 |- ((R Fr A /\ x e. A) -> -. xRx)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   = wceq 988   e. wcel 990  {cab 1499   =/= wne 1622  E.wrex 1684   i^i cin 2090   (_ wss 2091  (/)c0 2324  {csn 2454   class class class wbr 2669   Fr wfr 2970

This theorem is referenced by:  efrirr 2983  dfwe2 2990

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-fr 2972

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