Theorem dtru 3498

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Note that we may not substitute the same variable for both x and y (as indicated by the distinct variable requirement), for otherwise we would contradict stdpc6 1486. Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that x and y be distinct. Specifically, theorem cla4ev 2371 requires that x must not occur in the subexpression -. y = {(/)} in step 4 nor in the subexpression -. y = (/) in step 9. The proof verifier will require that x and y be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-16 1580, ax-ext 1865, or ax-sep 3438. See dtruALT 3517 for a shorter proof using these axioms.

Assertion

Ref	Expression
dtru	|- -. A.x x = y

Distinct variable group:   x,y

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		eeanv 1707	. . . . 5 |- (E.wE.z(x e. w /\ -. x e. z) <-> (E.w x e. w /\ E.z -. x e. z))
2		ax-pow 3481	. . . . . 6 |- E.wA.z(A.y(y e. z -> y e. x) -> z e. w)
3		id 73	. . . . . . . . 9 |- (y e. x -> y e. x)
4	3	ax-gen 1305	. . . . . . . 8 |- A.y(y e. x -> y e. x)
5		elequ2 1497	. . . . . . . . . . . 12 |- (z = x -> (y e. z <-> y e. x))
6	5	imbi1d 675	. . . . . . . . . . 11 |- (z = x -> ((y e. z -> y e. x) <-> (y e. x -> y e. x)))
7	6	albidv 1656	. . . . . . . . . 10 |- (z = x -> (A.y(y e. z -> y e. x) <-> A.y(y e. x -> y e. x)))
8		elequ1 1496	. . . . . . . . . 10 |- (z = x -> (z e. w <-> x e. w))
9	7, 8	imbi12d 688	. . . . . . . . 9 |- (z = x -> ((A.y(y e. z -> y e. x) -> z e. w) <-> (A.y(y e. x -> y e. x) -> x e. w)))
10	9	a4v 1649	. . . . . . . 8 |- (A.z(A.y(y e. z -> y e. x) -> z e. w) -> (A.y(y e. x -> y e. x) -> x e. w))
11	4, 10	mpi 55	. . . . . . 7 |- (A.z(A.y(y e. z -> y e. x) -> z e. w) -> x e. w)
12	11	eximi 1387	. . . . . 6 |- (E.wA.z(A.y(y e. z -> y e. x) -> z e. w) -> E.w x e. w)
13	2, 12	ax-mp 7	. . . . 5 |- E.w x e. w
14		ax-nul 3445	. . . . . 6 |- E.zA.x -. x e. z
15		ax-4 1319	. . . . . . 7 |- (A.x -. x e. z -> -. x e. z)
16	15	eximi 1387	. . . . . 6 |- (E.zA.x -. x e. z -> E.z -. x e. z)
17	14, 16	ax-mp 7	. . . . 5 |- E.z -. x e. z
18	1, 13, 17	mpbir2an 800	. . . 4 |- E.wE.z(x e. w /\ -. x e. z)
19		ax-14 1312	. . . . . . . 8 |- (w = z -> (x e. w -> x e. z))
20	19	com12 14	. . . . . . 7 |- (x e. w -> (w = z -> x e. z))
21	20	con3d 111	. . . . . 6 |- (x e. w -> (-. x e. z -> -. w = z))
22	21	imp 377	. . . . 5 |- ((x e. w /\ -. x e. z) -> -. w = z)
23	22	2eximi 1388	. . . 4 |- (E.wE.z(x e. w /\ -. x e. z) -> E.wE.z -. w = z)
24	18, 23	ax-mp 7	. . 3 |- E.wE.z -. w = z
25		equequ2 1495	. . . . . . 7 |- (z = y -> (w = z <-> w = y))
26	25	notbid 673	. . . . . 6 |- (z = y -> (-. w = z <-> -. w = y))
27		ax-8 1306	. . . . . . . 8 |- (x = w -> (x = y -> w = y))
28	27	con3d 111	. . . . . . 7 |- (x = w -> (-. w = y -> -. x = y))
29	28	a4imev 1650	. . . . . 6 |- (-. w = y -> E.x -. x = y)
30	26, 29	syl6bi 231	. . . . 5 |- (z = y -> (-. w = z -> E.x -. x = y))
31		ax-8 1306	. . . . . . . 8 |- (x = z -> (x = y -> z = y))
32	31	con3d 111	. . . . . . 7 |- (x = z -> (-. z = y -> -. x = y))
33	32	a4imev 1650	. . . . . 6 |- (-. z = y -> E.x -. x = y)
34	33	a1d 15	. . . . 5 |- (-. z = y -> (-. w = z -> E.x -. x = y))
35	30, 34	pm2.61i 140	. . . 4 |- (-. w = z -> E.x -. x = y)
36	35	19.23aivv 1675	. . 3 |- (E.wE.z -. w = z -> E.x -. x = y)
37	24, 36	ax-mp 7	. 2 |- E.x -. x = y
38		exnal 1385	. 2 |- (E.x -. x = y <-> -. A.x x = y)
39	37, 38	mpbi 206	1 |- -. A.x x = y

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem is referenced by:  ax16b 3499  eunex 3500  dtrucor 3518  dvdemo1 3520  zfcndpow 6120

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-nul 3445  ax-pow 3481

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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