Theorem sndisj 5077

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 5054	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
2		moeq 3653	. . 3 ⊢ ∃*𝑥 𝑥 = 𝑦
3		simpr 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4	3	elsnd 4585	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
5	4	equcomd 2021	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
6	5	moimi 2545	. . 3 ⊢ (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
7	2, 6	ax-mp 5	. 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})
8	1, 7	mpgbir 1801	1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}

Syntax hints:   ∧ wa 395   ∈ wcel 2114  ∃*wmo 2537  {csn 4567  Disj wdisj 5052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-rmo 3342  df-sn 4568  df-disj 5053

This theorem is referenced by:  0disj  5078  fnpreimac  32743  elrspunidl  33488  sibfof  34484  disjsnxp  45501  vonct  47121