Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funsneqopOLD Structured version   Visualization version   GIF version

Theorem funsneqopOLD 6670
 Description: Obsolete as of 15-Jul-2022. (Contributed by AV, 24-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
funsndifnop.a 𝐴 ∈ V
funsndifnop.b 𝐵 ∈ V
funsndifnop.g 𝐺 = {⟨𝐴, 𝐵⟩}
Assertion
Ref Expression
funsneqopOLD (𝐴 = 𝐵𝐺 ∈ (V × V))

Proof of Theorem funsneqopOLD
StepHypRef Expression
1 funsndifnop.a . . 3 𝐴 ∈ V
2 funsndifnop.b . . 3 𝐵 ∈ V
3 funsndifnop.g . . 3 𝐺 = {⟨𝐴, 𝐵⟩}
41, 2, 3funsneqopsnOLD 6669 . 2 (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)
5 snex 5130 . . 3 {𝐴} ∈ V
65, 5opelvv 5382 . 2 ⟨{𝐴}, {𝐴}⟩ ∈ (V × V)
74, 6syl6eqel 2915 1 (𝐴 = 𝐵𝐺 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3415  {csn 4398  ⟨cop 4404   × cxp 5341 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4937  df-xp 5349 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator