Theorem exneq 5403

Description: Given any set (the "𝑦 " in the statement), there exists a set not equal to it.

The same statement without disjoint variable condition is false, since we do not have ∃𝑥¬ 𝑥 = 𝑥. This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext 2734, ax-sep 5246, or ax-pow 5322 nor auxiliary logical axiom schemes ax-10 2175 to ax-13 2403. See dtruALT 5345 for a shorter proof using more axioms, and dtruALT2 5327 for a proof using ax-pow 5322 instead of ax-pr 5390. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2403. (Revised by BJ, 31-May-2019.) Avoid ax-8 2144. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2212. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5390 instead of ax-pow 5322. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5404. (Revised by BJ, 2-Jan-2025.)

Assertion

Ref	Expression
exneq	⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem exneq

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

Step	Hyp	Ref	Expression
1		exexneq 5402	. 2 ⊢ ∃𝑧∃𝑤 ¬ 𝑧 = 𝑤
2		equeuclr 2043	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑤))
3	2	con3d 152	. . . . 5 ⊢ (𝑤 = 𝑦 → (¬ 𝑧 = 𝑤 → ¬ 𝑧 = 𝑦))
4		ax7v1 2030	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
5	4	con3d 152	. . . . . 6 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
6	5	spimevw 2005	. . . . 5 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
7	3, 6	syl6 35	. . . 4 ⊢ (𝑤 = 𝑦 → (¬ 𝑧 = 𝑤 → ∃𝑥 ¬ 𝑥 = 𝑦))
8		ax7v1 2030	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦))
9	8	con3d 152	. . . . . 6 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
10	9	spimevw 2005	. . . . 5 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
11	10	a1d 25	. . . 4 ⊢ (¬ 𝑤 = 𝑦 → (¬ 𝑧 = 𝑤 → ∃𝑥 ¬ 𝑥 = 𝑦))
12	7, 11	pm2.61i 183	. . 3 ⊢ (¬ 𝑧 = 𝑤 → ∃𝑥 ¬ 𝑥 = 𝑦)
13	12	exlimivv 1952	. 2 ⊢ (∃𝑧∃𝑤 ¬ 𝑧 = 𝑤 → ∃𝑥 ¬ 𝑥 = 𝑦)
14	1, 13	ax-mp 5	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800

This theorem is referenced by:  dtru  5404