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Mirrors > Home > MPE Home > Th. List > 3xpexg | Structured version Visualization version GIF version |
Description: The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.) |
Ref | Expression |
---|---|
3xpexg | ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpexg 7741 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑉 ∈ 𝑊) → (𝑉 × 𝑉) ∈ V) | |
2 | 1 | anidms 566 | . 2 ⊢ (𝑉 ∈ 𝑊 → (𝑉 × 𝑉) ∈ V) |
3 | xpexg 7741 | . 2 ⊢ (((𝑉 × 𝑉) ∈ V ∧ 𝑉 ∈ 𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V) | |
4 | 2, 3 | mpancom 685 | 1 ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3473 × cxp 5674 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-opab 5211 df-xp 5682 df-rel 5683 |
This theorem is referenced by: (None) |
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