Theorem dtruarb 4177

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 4543 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 4164	. . 3 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-nul 4115	. . . 4 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦
3		sp 1504	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑦)
4	2, 3	eximii 1595	. . 3 ⊢ ∃𝑦 ¬ 𝑧 ∈ 𝑦
5		eeanv 1925	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) ↔ (∃𝑥 𝑧 ∈ 𝑥 ∧ ∃𝑦 ¬ 𝑧 ∈ 𝑦))
6	1, 4, 5	mpbir2an 937	. 2 ⊢ ∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)
7		nelneq2 2272	. . 3 ⊢ ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ¬ 𝑥 = 𝑦)
8	7	2eximi 1594	. 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 103  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166

This theorem is referenced by: (None)