Theorem difprsnss 3847

Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
difprsnss	⊢ ({A, B} ∖ {A}) ⊆ {B}

Proof of Theorem difprsnss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . 5 ⊢ x ∈ V
2	1	elpr 3752	. . . 4 ⊢ (x ∈ {A, B} ↔ (x = A ∨ x = B))
3		elsn 3749	. . . . 5 ⊢ (x ∈ {A} ↔ x = A)
4	3	notbii 287	. . . 4 ⊢ (¬ x ∈ {A} ↔ ¬ x = A)
5		biorf 394	. . . . 5 ⊢ (¬ x = A → (x = B ↔ (x = A ∨ x = B)))
6	5	biimparc 473	. . . 4 ⊢ (((x = A ∨ x = B) ∧ ¬ x = A) → x = B)
7	2, 4, 6	syl2anb 465	. . 3 ⊢ ((x ∈ {A, B} ∧ ¬ x ∈ {A}) → x = B)
8		eldif 3222	. . 3 ⊢ (x ∈ ({A, B} ∖ {A}) ↔ (x ∈ {A, B} ∧ ¬ x ∈ {A}))
9		elsn 3749	. . 3 ⊢ (x ∈ {B} ↔ x = B)
10	7, 8, 9	3imtr4i 257	. 2 ⊢ (x ∈ ({A, B} ∖ {A}) → x ∈ {B})
11	10	ssriv 3278	1 ⊢ ({A, B} ∖ {A}) ⊆ {B}

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ⊆ wss 3258  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-sn 3742  df-pr 3743

This theorem is referenced by: (None)