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Mirrors > Home > MPE Home > Th. List > constr3lem1 | Structured version Visualization version GIF version |
Description: Lemma for constr3trl 26187 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.) |
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〉}) |
Ref | Expression |
---|---|
constr3lem1 | ⊢ (𝐹 ∈ V ∧ 𝑃 ∈ V) |
Step | Hyp | Ref | 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 | |
3 | 1, 2 | eqeltri 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 | |
7 | 5, 6 | unex 6854 | . . 3 ⊢ ({〈0, 𝐴〉, 〈1, 𝐵〉} ∪ {〈2, 𝐶〉, 〈3, 𝐴〉}) ∈ V |
8 | 4, 7 | eqeltri 2684 | . 2 ⊢ 𝑃 ∈ V |
9 | 3, 8 | pm3.2i 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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