Theorem difprsnss 4775

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 3463	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elpr 4626	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
3		velsn 4617	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4	3	notbii 320	. . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴)
5		biorf 936	. . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
6	5	biimparc 479	. . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵)
7	2, 4, 6	syl2anb 598	. . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵)
8		eldif 3936	. . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}))
9		velsn 4617	. . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
10	7, 8, 9	3imtr4i 292	. 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵})
11	10	ssriv 3962	1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ∖ cdif 3923   ⊆ wss 3926  {csn 4601  {cpr 4603

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-sn 4602  df-pr 4604

This theorem is referenced by:  en2other2  10023  pmtrprfv  19434  itg11  25644