| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| 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 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝑌 | |
| 4 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐴 | |
| 5 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐵 | |
| 6 | nfcv 2904 | . 2 ⊢ Ⅎ𝑥𝐶 | |
| 7 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐶 | |
| 8 | nfcv 2904 | . 2 ⊢ Ⅎ𝑥𝐷 | |
| 9 | nfcv 2904 | . 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 43650 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3950 ∪ ciun 4989 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-ss 3967 df-iun 4991 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |