Theorem dtru 2767

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 1125. Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that x and y be distinct. Specifically, theorem cla4ev 1865 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 2743 for a version proved without using ax-16 1208, ax-ext 1457, or ax-sep 2698.

Assertion

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

Distinct variable group:   x,y

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		0inp0 2733	. . . 4 |- (y = (/) -> -. y = {(/)})
2		p0ex 2765	. . . . 5 |- {(/)} e. V
3		eqeq2 1481	. . . . . 6 |- (x = {(/)} -> (y = x <-> y = {(/)}))
4	3	negbid 610	. . . . 5 |- (x = {(/)} -> (-. y = x <-> -. y = {(/)}))
5	2, 4	cla4ev 1865	. . . 4 |- (-. y = {(/)} -> E.x -. y = x)
6	1, 5	syl 10	. . 3 |- (y = (/) -> E.x -. y = x)
7		0ex 2706	. . . 4 |- (/) e. V
8		eqeq2 1481	. . . . 5 |- (x = (/) -> (y = x <-> y = (/)))
9	8	negbid 610	. . . 4 |- (x = (/) -> (-. y = x <-> -. y = (/)))
10	7, 9	cla4ev 1865	. . 3 |- (-. y = (/) -> E.x -. y = x)
11	6, 10	pm2.61i 126	. 2 |- E.x -. y = x
12		exnal 1036	. . 3 |- (E.x -. y = x <-> -. A.x y = x)
13		eqcom 1474	. . . . 5 |- (y = x <-> x = y)
14	13	albii 997	. . . 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 952   = wceq 954  E.wex 978  (/)c0 2276  {csn 2405

This theorem is referenced by:  dtrucor 2768  dvdemo1 2770  zfcndpow 4948

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-nul 2705  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-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

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