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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 4575 | . . 3 ⊢ 〈𝐷, 𝐻, 𝐴〉 = 〈〈𝐷, 𝐻〉, 𝐴〉 | |
2 | mpstssv.p | . . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇) | |
3 | 2 | mpstssv 32786 | . . . 4 ⊢ 𝑃 ⊆ ((V × V) × V) |
4 | 3 | sseli 3962 | . . 3 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈𝐷, 𝐻, 𝐴〉 ∈ ((V × V) × V)) |
5 | 1, 4 | eqeltrrid 2918 | . 2 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V)) |
6 | opelxp 5590 | . . . 4 ⊢ (〈𝐷, 𝐻〉 ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V)) | |
7 | 6 | anbi1i 625 | . . 3 ⊢ ((〈𝐷, 𝐻〉 ∈ (V × V) ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V)) |
8 | opelxp 5590 | . . 3 ⊢ (〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V) ↔ (〈𝐷, 𝐻〉 ∈ (V × V) ∧ 𝐴 ∈ V)) | |
9 | df-3an 1085 | . . 3 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V)) | |
10 | 7, 8, 9 | 3bitr4i 305 | . 2 ⊢ (〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
11 | 5, 10 | sylib 220 | 1 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 〈cotp 4574 × cxp 5552 ‘cfv 6354 mPreStcmpst 32720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-ot 4575 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-mpst 32740 |
This theorem is referenced by: elmsta 32795 mclsax 32816 |
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