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Theorem difprsnss 4692
 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.
StepHypRef Expression
1 vex 3444 . . . . 5 𝑥 ∈ V
21elpr 4548 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 4541 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
43notbii 323 . . . 4 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴)
5 biorf 934 . . . . 5 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
65biimparc 483 . . . 4 (((𝑥 = 𝐴𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵)
72, 4, 6syl2anb 600 . . 3 ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵)
8 eldif 3891 . . 3 (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}))
9 velsn 4541 . . 3 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
107, 8, 93imtr4i 295 . 2 (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵})
1110ssriv 3919 1 ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ⊆ wss 3881  {csn 4525  {cpr 4527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528 This theorem is referenced by:  en2other2  9423  pmtrprfv  18577  itg11  24305
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