Theorem dtru 4994

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 2098.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2728 or ax-sep 4921. See dtruALT 5036 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 4984	. . . 4 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		ax-nul 4929	. . . . 5 ⊢ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧
3		sp 2188	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧)
4	2, 3	eximii 1901	. . . 4 ⊢ ∃𝑧 ¬ 𝑥 ∈ 𝑧
5		eeanv 2315	. . . 4 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) ↔ (∃𝑤 𝑥 ∈ 𝑤 ∧ ∃𝑧 ¬ 𝑥 ∈ 𝑧))
6	1, 4, 5	mpbir2an 993	. . 3 ⊢ ∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧)
7		ax9 2140	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧))
8	7	com12 32	. . . . 5 ⊢ (𝑥 ∈ 𝑤 → (𝑤 = 𝑧 → 𝑥 ∈ 𝑧))
9	8	con3dimp 456	. . . 4 ⊢ ((𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ¬ 𝑤 = 𝑧)
10	9	2eximi 1900	. . 3 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧)
11		equequ2 2096	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑤 = 𝑦))
12	11	notbid 307	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
13		ax7 2086	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦))
14	13	con3d 148	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
15	14	spimev 2392	. . . . . 6 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
16	12, 15	syl6bi 243	. . . . 5 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
17		ax7 2086	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
18	17	con3d 148	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
19	18	spimev 2392	. . . . . 6 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
20	19	a1d 25	. . . . 5 ⊢ (¬ 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
21	16, 20	pm2.61i 176	. . . 4 ⊢ (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
22	21	exlimivv 1997	. . 3 ⊢ (∃𝑤∃𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
23	6, 10, 22	mp2b 10	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
24		exnal 1891	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
25	23, 24	mpbi 220	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1618  ∃wex 1841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-nul 4929  ax-pow 4980

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847

This theorem is referenced by:  axc16b  4995  eunex  4996  nfnid  5034  dtrucor  5037  dvdemo1  5039  brprcneu  6333  zfcndpow  9601