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Theorem constr3lem1 26173
 Description: Lemma for constr3trl 26187 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
Assertion
Ref Expression
constr3lem1 (𝐹 ∈ V ∧ 𝑃 ∈ V)

Proof of Theorem constr3lem1
StepHypRef Expression
1 constr3cycl.f . . 3 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
2 tpex 6855 . . 3 {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩} ∈ V
31, 2eqeltri 2684 . 2 𝐹 ∈ V
4 constr3cycl.p . . 3 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
5 prex 4836 . . . 4 {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ V
6 prex 4836 . . . 4 {⟨2, 𝐶⟩, ⟨3, 𝐴⟩} ∈ V
75, 6unex 6854 . . 3 ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) ∈ V
84, 7eqeltri 2684 . 2 𝑃 ∈ V
93, 8pm3.2i 470 1 (𝐹 ∈ V ∧ 𝑃 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  {cpr 4127  {ctp 4129  ⟨cop 4131  ◡ccnv 5037  ‘cfv 5804  0cc0 9815  1c1 9816  2c2 10947  3c3 10948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-tp 4130  df-uni 4373 This theorem is referenced by:  constr3trl  26187  constr3pth  26188  constr3cycl  26189
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