Theorem dtruarb 3942

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 4283 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 3931	. . 3 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-nul 3883	. . . 4 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦
3		sp 1401	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑦)
4	2, 3	eximii 1493	. . 3 ⊢ ∃𝑦 ¬ 𝑧 ∈ 𝑦
5		eeanv 1807	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) ↔ (∃𝑥 𝑧 ∈ 𝑥 ∧ ∃𝑦 ¬ 𝑧 ∈ 𝑦))
6	1, 4, 5	mpbir2an 849	. 2 ⊢ ∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)
7		nelneq2 2139	. . 3 ⊢ ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ¬ 𝑥 = 𝑦)
8	7	2eximi 1492	. 2 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦)
9	6, 8	ax-mp 7	1 ⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 97  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022  ax-nul 3883  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036

This theorem is referenced by: (None)