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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpstrcl | Structured version Visualization version GIF version |
Description: The elements of a pre-statement are sets. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mpstssv.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
Ref | Expression |
---|---|
mpstrcl | ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4134 | . . 3 ⊢ 〈𝐷, 𝐻, 𝐴〉 = 〈〈𝐷, 𝐻〉, 𝐴〉 | |
2 | mpstssv.p | . . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇) | |
3 | 2 | mpstssv 30690 | . . . 4 ⊢ 𝑃 ⊆ ((V × V) × V) |
4 | 3 | sseli 3564 | . . 3 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈𝐷, 𝐻, 𝐴〉 ∈ ((V × V) × V)) |
5 | 1, 4 | syl5eqelr 2693 | . 2 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V)) |
6 | opelxp 5070 | . . . 4 ⊢ (〈𝐷, 𝐻〉 ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V)) | |
7 | 6 | anbi1i 727 | . . 3 ⊢ ((〈𝐷, 𝐻〉 ∈ (V × V) ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V)) |
8 | opelxp 5070 | . . 3 ⊢ (〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V) ↔ (〈𝐷, 𝐻〉 ∈ (V × V) ∧ 𝐴 ∈ V)) | |
9 | df-3an 1033 | . . 3 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V)) | |
10 | 7, 8, 9 | 3bitr4i 291 | . 2 ⊢ (〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
11 | 5, 10 | sylib 207 | 1 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 〈cotp 4133 × cxp 5036 ‘cfv 5804 mPreStcmpst 30624 |
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-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-mpst 30644 |
This theorem is referenced by: elmsta 30699 mclsax 30720 |
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