Theorem difprsnss 4762

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 3458	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elpr 4609	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
3		velsn 4600	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4	3	notbii 323	. . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴)
5		biorf 950	. . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
6	5	biimparc 485	. . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵)
7	2, 4, 6	syl2anb 610	. . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵)
8		eldif 4052	. . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}))
9		velsn 4600	. . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
10	7, 8, 9	3imtr4i 295	. 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵})
11	10	ssriv 3917	1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  {cpr 4590

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-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-sn 4589  df-pr 4591

This theorem is used by:  en2other2  10118  ex-chn1  18843  pmtrprfv  19700  itg11  26045