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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 4534 | . . 3 ⊢ 〈𝐷, 𝐻, 𝐴〉 = 〈〈𝐷, 𝐻〉, 𝐴〉 | |
2 | mpstssv.p | . . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇) | |
3 | 2 | mpstssv 32899 | . . . 4 ⊢ 𝑃 ⊆ ((V × V) × V) |
4 | 3 | sseli 3911 | . . 3 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈𝐷, 𝐻, 𝐴〉 ∈ ((V × V) × V)) |
5 | 1, 4 | eqeltrrid 2895 | . 2 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V)) |
6 | opelxp 5555 | . . . 4 ⊢ (〈𝐷, 𝐻〉 ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V)) | |
7 | 6 | anbi1i 626 | . . 3 ⊢ ((〈𝐷, 𝐻〉 ∈ (V × V) ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V)) |
8 | opelxp 5555 | . . 3 ⊢ (〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V) ↔ (〈𝐷, 𝐻〉 ∈ (V × V) ∧ 𝐴 ∈ V)) | |
9 | df-3an 1086 | . . 3 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V)) | |
10 | 7, 8, 9 | 3bitr4i 306 | . 2 ⊢ (〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
11 | 5, 10 | sylib 221 | 1 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 〈cotp 4533 × cxp 5517 ‘cfv 6324 mPreStcmpst 32833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-mpst 32853 |
This theorem is referenced by: elmsta 32908 mclsax 32929 |
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