Theorem dtru 2778

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 1129. Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that x and y be distinct. Specifically, theorem cla4ev 1872 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.

See dtruALT 2754 for a version proved without using ax-16 1212, ax-ext 1462, or ax-sep 2708.

Assertion

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

Distinct variable group:   x,y

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		0inp0 2743	. . . 4 |- (y = (/) -> -. y = {(/)})
2		p0ex 2776	. . . . 5 |- {(/)} e. V
3		eqeq2 1487	. . . . . 6 |- (x = {(/)} -> (y = x <-> y = {(/)}))
4	3	negbid 613	. . . . 5 |- (x = {(/)} -> (-. y = x <-> -. y = {(/)}))
5	2, 4	cla4ev 1872	. . . 4 |- (-. y = {(/)} -> E.x -. y = x)
6	1, 5	syl 10	. . 3 |- (y = (/) -> E.x -. y = x)
7		0ex 2716	. . . 4 |- (/) e. V
8		eqeq2 1487	. . . . 5 |- (x = (/) -> (y = x <-> y = (/)))
9	8	negbid 613	. . . 4 |- (x = (/) -> (-. y = x <-> -. y = (/)))
10	7, 9	cla4ev 1872	. . 3 |- (-. y = (/) -> E.x -. y = x)
11	6, 10	pm2.61i 126	. 2 |- E.x -. y = x
12		exnal 1040	. . 3 |- (E.x -. y = x <-> -. A.x y = x)
13		eqcom 1480	. . . . 5 |- (y = x <-> x = y)
14	13	albii 1001	. . . 4 |- (A.x y = x <-> A.x x = y)
15	14	negbii 187	. . 3 |- (-. A.x y = x <-> -. A.x x = y)
16	12, 15	bitr 173	. 2 |- (E.x -. y = x <-> -. A.x x = y)
17	11, 16	mpbi 189	1 |- -. A.x x = y

Syntax hints:  -. wn 2  A.wal 956   = wceq 958  E.wex 982  (/)c0 2283  {csn 2413

This theorem is referenced by:  dtrucor 2779  dvdemo1 2781  zfcndpow 4980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417

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