Theorem ovsn 48891

Description: The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)

Hypothesis

Ref	Expression
ovsn.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
ovsn	⊢ (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶

Proof of Theorem ovsn

Step	Hyp	Ref	Expression
1		ovsn.1	. 2 ⊢ 𝐶 ∈ V
2		ovsng 48889	. 2 ⊢ (𝐶 ∈ V → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)
3	1, 2	ax-mp 5	1 ⊢ (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4571  ⟨cop 4577  (class class class)co 7341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344

This theorem is referenced by: (None)