Theorem sndisj 5095

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 5072	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
2		moeq 3665	. . 3 ⊢ ∃*𝑥 𝑥 = 𝑦
3		simpr 490	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4	3	elsnd 4602	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
5	4	equcomd 2052	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
6	5	moimi 2570	. . 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 1832	1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2145  ∃*wmo 2562  {csn 4584  Disj wdisj 5070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-rmo 3365  df-sn 4585  df-disj 5071

This theorem is used by:  0disj  5096  fnpreimac  33143  elrspunidl  33856  sibfof  34851  disjsnxp  45904  vonct  47521