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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 4578 | . . 3 ⊢ 〈𝐷, 𝐻, 𝐴〉 = 〈〈𝐷, 𝐻〉, 𝐴〉 | |
2 | mpstssv.p | . . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇) | |
3 | 2 | mpstssv 32788 | . . . 4 ⊢ 𝑃 ⊆ ((V × V) × V) |
4 | 3 | sseli 3965 | . . 3 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈𝐷, 𝐻, 𝐴〉 ∈ ((V × V) × V)) |
5 | 1, 4 | eqeltrrid 2920 | . 2 ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → 〈〈𝐷, 𝐻〉, 𝐴〉 ∈ ((V × V) × V)) |
6 | opelxp 5593 | . . . 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 5593 | . . 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 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 〈cotp 4577 × cxp 5555 ‘cfv 6357 mPreStcmpst 32722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-mpst 32742 |
This theorem is referenced by: elmsta 32797 mclsax 32818 |
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