Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sprid Structured version   Visualization version   GIF version

Theorem sprid 47466
Description: Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
sprid {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Proof of Theorem sprid
StepHypRef Expression
1 rexv 3508 . . 3 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏})
2 rexv 3508 . . . 4 (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏})
32exbii 1847 . . 3 (∃𝑎𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
41, 3bitri 275 . 2 (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
54abbii 2808 1 {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wex 1778  {cab 2713  wrex 3069  Vcvv 3479  {cpr 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070  df-v 3481
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator