Theorem sndisj 5044

Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
sndisj	⊢ Disj 𝑥 ∈ 𝐴 {𝑥}

Proof of Theorem sndisj

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 5020	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
2		moeq 3620	. . 3 ⊢ ∃*𝑥 𝑥 = 𝑦
3		simpr 488	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4		velsn 4557	. . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
5	3, 4	sylib 221	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
6	5	equcomd 2027	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
7	6	moimi 2544	. . 3 ⊢ (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
8	2, 7	ax-mp 5	. 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})
9	1, 8	mpgbir 1807	1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}

Syntax hints:   ∧ wa 399   ∈ wcel 2110  ∃*wmo 2537  {csn 4541  Disj wdisj 5018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2816  df-rmo 3069  df-v 3410  df-sn 4542  df-disj 5019

This theorem is referenced by:  0disj  5045  fnpreimac  30728  elrspunidl  31320  sibfof  32019  disjsnxp  42291  vonct  43906