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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-unrab | Structured version Visualization version GIF version |
Description: Generalization of unrab 4300. Equality need not hold. (Contributed by BJ, 21-Apr-2019.) |
Ref | Expression |
---|---|
bj-unrab | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4167 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | rabss2 4070 | . . . 4 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑}) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑} |
4 | orc 864 | . . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝜑 → (𝜑 ∨ 𝜓))) |
6 | 5 | ss2rabi 4069 | . . 3 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
7 | 3, 6 | sstri 3986 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
8 | ssun2 4168 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
9 | rabss2 4070 | . . . 4 ⊢ (𝐵 ⊆ (𝐴 ∪ 𝐵) → {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓}) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓} |
11 | olc 865 | . . . . 5 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝜓 → (𝜑 ∨ 𝜓))) |
13 | 12 | ss2rabi 4069 | . . 3 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
14 | 10, 13 | sstri 3986 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
15 | 7, 14 | unssi 4180 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 ∈ wcel 2098 {crab 3426 ∪ cun 3941 ⊆ wss 3943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rab 3427 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 |
This theorem is referenced by: (None) |
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