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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 1910 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1910 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2901 | . 2 ⊢ Ⅎ𝑦𝑌 | |
4 | nfcv 2901 | . 2 ⊢ Ⅎ𝑦𝐴 | |
5 | nfcv 2901 | . 2 ⊢ Ⅎ𝑦𝐵 | |
6 | nfcv 2901 | . 2 ⊢ Ⅎ𝑥𝐶 | |
7 | nfcv 2901 | . 2 ⊢ Ⅎ𝑦𝐶 | |
8 | nfcv 2901 | . 2 ⊢ Ⅎ𝑥𝐷 | |
9 | nfcv 2901 | . 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 43607 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ⊆ wss 3963 ∪ ciun 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-v 3479 df-ss 3980 df-iun 5000 |
This theorem is referenced by: (None) |
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