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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-unrab | Structured version Visualization version GIF version |
Description: Generalization of unrab 4271. 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 4145 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | rabss2 4051 | . . . 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 861 | . . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝜑 → (𝜑 ∨ 𝜓))) |
6 | 5 | ss2rabi 4050 | . . 3 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
7 | 3, 6 | sstri 3973 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
8 | ssun2 4146 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
9 | rabss2 4051 | . . . 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 862 | . . . . 5 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝜓 → (𝜑 ∨ 𝜓))) |
13 | 12 | ss2rabi 4050 | . . 3 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
14 | 10, 13 | sstri 3973 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
15 | 7, 14 | unssi 4158 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 841 ∈ wcel 2105 {crab 3139 ∪ cun 3931 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-un 3938 df-in 3940 df-ss 3949 |
This theorem is referenced by: (None) |
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