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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 7462 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑉 ∈ 𝑊) → (𝑉 × 𝑉) ∈ V) | |
2 | 1 | anidms 567 | . 2 ⊢ (𝑉 ∈ 𝑊 → (𝑉 × 𝑉) ∈ V) |
3 | xpexg 7462 | . 2 ⊢ (((𝑉 × 𝑉) ∈ V ∧ 𝑉 ∈ 𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V) | |
4 | 2, 3 | mpancom 684 | 1 ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: (None) |
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