Theorem difprsnss 4801

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 3476	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elpr 4650	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
3		velsn 4643	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4	3	notbii 319	. . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴)
5		biorf 933	. . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
6	5	biimparc 478	. . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵)
7	2, 4, 6	syl2anb 596	. . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵)
8		eldif 3957	. . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}))
9		velsn 4643	. . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
10	7, 8, 9	3imtr4i 291	. 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵})
11	10	ssriv 3985	1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}

Syntax hints:  ¬ wn 3   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   ∖ cdif 3944   ⊆ wss 3947  {csn 4627  {cpr 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630

This theorem is referenced by:  en2other2  10006  pmtrprfv  19362  itg11  25440