Theorem disjxsn 4086

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 4066	. 2 ⊢ (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵))
2		moeq 2981	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
3		elsni 3687	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
4	3	adantr 276	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐴)
5	4	moimi 2145	. . 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 1501	1 ⊢ Disj 𝑥 ∈ {𝐴}𝐵

Syntax hints:   ∧ wa 104   = wceq 1397  ∃*wmo 2080   ∈ wcel 2202  {csn 3669  Disj wdisj 4064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rmo 2518  df-v 2804  df-sn 3675  df-disj 4065

This theorem is referenced by:  disjx0  4087