Theorem disjxsn 5100

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 5075	. 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵))
2		moeq 3668	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
3		elsni 4601	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
4	3	adantr 486	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴)
5	4	moimi 2573	. . 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 1832	1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃*wmo 2565  {csn 4588  Disj wdisj 5073

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-rmo 3369  df-sn 4589  df-disj 5074

This theorem is used by:  disjx0  5101  disjdifprg  33241  rossros  34883  meadjun  47677