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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 450 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶)) |
4 | 1, 3 | ralrimi 2957 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
5 | ss2ixp 7921 | . 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 384 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 Xcixp 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-in 3581 df-ss 3588 df-ixp 7909 |
This theorem is referenced by: ioosshoi 40883 iinhoiicclem 40887 iinhoiicc 40888 iunhoiioo 40890 vonioolem2 40895 vonicclem2 40898 |
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