Theorem disjxsn 5101

Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjxsn	⊢ Disj 𝑥 ∈ {𝐴}𝐵

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjxsn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 5076	. 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵))
2		moeq 3678	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
3		elsni 4606	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
4	3	adantr 480	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴)
5	4	moimi 2538	. . 3 ⊢ (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵))
6	2, 5	ax-mp 5	. 2 ⊢ ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵)
7	1, 6	mpgbir 1799	1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  {csn 4589  Disj wdisj 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-rmo 3354  df-sn 4590  df-disj 5075

This theorem is referenced by:  disjx0  5102  disjdifprg  32504  rossros  34170  meadjun  46460