Theorem sndisj 4574

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 4555	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}))
2		moeq 3349	. . 3 ⊢ ∃*𝑥 𝑥 = 𝑦
3		simpr 476	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4		velsn 4141	. . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
5	3, 4	sylib 207	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
6	5	equcomd 1933	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
7	6	moimi 2508	. . 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 1717	1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}

Syntax hints:   ∧ wa 383   ∈ wcel 1977  ∃*wmo 2459  {csn 4125  Disj wdisj 4553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904  df-v 3175  df-sn 4126  df-disj 4554

This theorem is referenced by:  0disj  4575  sibfof  29729  disjsnxp  38265  vonct  39584