Theorem disjxsn 5063

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 5037	. 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵))
2		moeq 3637	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
3		elsni 4575	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
4	3	adantr 480	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴)
5	4	moimi 2545	. . 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 1803	1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  {csn 4558  Disj wdisj 5035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-rmo 3071  df-sn 4559  df-disj 5036

This theorem is referenced by:  disjx0  5064  disjdifprg  30815  rossros  32048  meadjun  43890