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Theorem difprsnss 4795
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 3470 . . . . 5 𝑥 ∈ V
21elpr 4644 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 4637 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
43notbii 320 . . . 4 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴)
5 biorf 933 . . . . 5 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
65biimparc 479 . . . 4 (((𝑥 = 𝐴𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵)
72, 4, 6syl2anb 597 . . 3 ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵)
8 eldif 3951 . . 3 (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}))
9 velsn 4637 . . 3 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
107, 8, 93imtr4i 292 . 2 (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵})
1110ssriv 3979 1 ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 844   = wceq 1533  wcel 2098  cdif 3938  wss 3941  {csn 4621  {cpr 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624
This theorem is referenced by:  en2other2  10001  pmtrprfv  19365  itg11  25544
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