Theorem dtru 4633

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 1906.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2485 or ax-sep 4558. See dtruALT 4673 for a shorter proof using these axioms.

The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.)

Assertion

Ref	Expression
dtru	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtru

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		el 4623	. . . 4 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		ax-nul 4567	. . . . 5 ⊢ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧
3		sp 1990	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧)
4	2, 3	eximii 1740	. . . 4 ⊢ ∃𝑧 ¬ 𝑥 ∈ 𝑧
5		eeanv 2126	. . . 4 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) ↔ (∃𝑤 𝑥 ∈ 𝑤 ∧ ∃𝑧 ¬ 𝑥 ∈ 𝑧))
6	1, 4, 5	mpbir2an 947	. . 3 ⊢ ∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧)
7		ax9 1950	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧))
8	7	com12 32	. . . . 5 ⊢ (𝑥 ∈ 𝑤 → (𝑤 = 𝑧 → 𝑥 ∈ 𝑧))
9	8	con3dimp 450	. . . 4 ⊢ ((𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ¬ 𝑤 = 𝑧)
10	9	2eximi 1739	. . 3 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧)
11		equequ2 1902	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑤 = 𝑦))
12	11	notbid 303	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
13		ax7 1892	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦))
14	13	con3d 142	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
15	14	spimev 2150	. . . . . 6 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
16	12, 15	syl6bi 238	. . . . 5 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
17		ax7 1892	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
18	17	con3d 142	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
19	18	spimev 2150	. . . . . 6 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
20	19	a1d 25	. . . . 5 ⊢ (¬ 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
21	16, 20	pm2.61i 171	. . . 4 ⊢ (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
22	21	exlimivv 1809	. . 3 ⊢ (∃𝑤∃𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
23	6, 10, 22	mp2b 10	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
24		exnal 1730	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
25	23, 24	mpbi 215	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 378  ∀wal 1466  ∃wex 1692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-nul 4567  ax-pow 4619

This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697

This theorem is referenced by:  axc16b  4634  eunex  4635  nfnid  4670  dtrucor  4674  dvdemo1  4676  brprcneu  5920  zfcndpow  9126