Theorem dtruarb 4193

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4560 in which we are given a set 𝑦 and go from there to a set 𝑥 which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)

Assertion

Ref	Expression
dtruarb	⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruarb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		el 4180	. . 3 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-nul 4131	. . . 4 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦
3		sp 1511	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑦)
4	2, 3	eximii 1602	. . 3 ⊢ ∃𝑦 ¬ 𝑧 ∈ 𝑦
5		eeanv 1932	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) ↔ (∃𝑥 𝑧 ∈ 𝑥 ∧ ∃𝑦 ¬ 𝑧 ∈ 𝑦))
6	1, 4, 5	mpbir2an 942	. 2 ⊢ ∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)
7		nelneq2 2279	. . 3 ⊢ ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ¬ 𝑥 = 𝑦)
8	7	2eximi 1601	. 2 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦)
9	6, 8	ax-mp 5	1 ⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 104  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173

This theorem is referenced by: (None)