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Theorem difprsnss 4799
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 3484 . . . . 5 𝑥 ∈ V
21elpr 4650 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 4642 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
43notbii 320 . . . 4 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴)
5 biorf 937 . . . . 5 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
65biimparc 479 . . . 4 (((𝑥 = 𝐴𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵)
72, 4, 6syl2anb 598 . . 3 ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵)
8 eldif 3961 . . 3 (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}))
9 velsn 4642 . . 3 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
107, 8, 93imtr4i 292 . 2 (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵})
1110ssriv 3987 1 ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 848   = wceq 1540  wcel 2108  cdif 3948  wss 3951  {csn 4626  {cpr 4628
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 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629
This theorem is referenced by:  en2other2  10049  pmtrprfv  19471  itg11  25726
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