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Mirrors > Home > MPE Home > Th. List > Mathboxes > ss2iundv | Structured version Visualization version GIF version |
Description: Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.) |
Ref | Expression |
---|---|
ss2iundv.el | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶) |
ss2iundv.sub | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) |
ss2iundv.ss | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺) |
Ref | Expression |
---|---|
ss2iundv | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝑌 | |
4 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝐴 | |
5 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝐵 | |
6 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐶 | |
7 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝐶 | |
8 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐷 | |
9 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝐺 | |
10 | ss2iundv.el | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶) | |
11 | ss2iundv.sub | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) | |
12 | ss2iundv.ss | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ss2iundf 36970 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ ciun 4455 |
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-3an 1033 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-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-iun 4457 |
This theorem is referenced by: (None) |
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