Theorem dtru 2768

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 2744 for a version proved without using ax-16 1208, ax-ext 1457, or ax-sep 2699.

Assertion

Ref	Expression
dtru	⊢ ¬ ∀x x = y

Distinct variable group:   x,y

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		0inp0 2734	. . . 4 ⊢ (y = ∅ → ¬ y = {∅})
2		p0ex 2766	. . . . 5 ⊢ {∅} ∈ 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 = {∅} → ∃x ¬ y = x)
6	1, 5	syl 10	. . 3 ⊢ (y = ∅ → ∃x ¬ y = x)
7		0ex 2707	. . . 4 ⊢ ∅ ∈ 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 = ∅ → ∃x ¬ y = x)
11	6, 10	pm2.61i 126	. 2 ⊢ ∃x ¬ y = x
12		exnal 1036	. . 3 ⊢ (∃x ¬ y = x ↔ ¬ ∀x y = x)
13		eqcom 1474	. . . . 5 ⊢ (y = x ↔ x = y)
14	13	albii 997	. . . 4 ⊢ (∀x y = x ↔ ∀x x = y)
15	14	negbii 187	. . 3 ⊢ (¬ ∀x y = x ↔ ¬ ∀x x = y)
16	12, 15	bitr 173	. 2 ⊢ (∃x ¬ y = x ↔ ¬ ∀x x = y)
17	11, 16	mpbi 189	1 ⊢ ¬ ∀x x = y

Syntax hints:  ¬ wn 2  ∀wal 952   = wceq 954  ∃wex 978  ∅c0 2276  {csn 2405

This theorem is referenced by:  dtrucor 2769  dvdemo1 2771  zfcndpow 4960

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 2699  ax-nul 2706  ax-pow 2738

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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