Theorem sndisj 5140

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 5117	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
2		moeq 3716	. . 3 ⊢ ∃*𝑥 𝑥 = 𝑦
3		simpr 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4		velsn 4647	. . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
5	3, 4	sylib 218	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
6	5	equcomd 2016	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
7	6	moimi 2543	. . 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 1796	1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}

Syntax hints:   ∧ wa 395   ∈ wcel 2106  ∃*wmo 2536  {csn 4631  Disj wdisj 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-rmo 3378  df-v 3480  df-sn 4632  df-disj 5116

This theorem is referenced by:  0disj  5141  fnpreimac  32688  elrspunidl  33436  sibfof  34322  disjsnxp  45010  vonct  46649