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Mirrors > Home > MPE Home > Th. List > Mathboxes > ixpssixp | Structured version Visualization version GIF version |
Description: Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
ixpssixp.1 | ⊢ Ⅎ𝑥𝜑 |
ixpssixp.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
ixpssixp | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpssixp.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ixpssixp.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) | |
3 | 2 | ex 449 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶)) |
4 | 1, 3 | ralrimi 2940 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
5 | ss2ixp 7807 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 Xcixp 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-in 3547 df-ss 3554 df-ixp 7795 |
This theorem is referenced by: ioosshoi 39560 iinhoiicclem 39564 iinhoiicc 39565 iunhoiioo 39567 vonioolem2 39572 vonicclem2 39575 |
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