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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inrab2 | Structured version Visualization version GIF version |
Description: Shorter proof of inrab 4241. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-inrab2 | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-inrab 35101 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ (𝐴 ∩ 𝐴) ∣ (𝜑 ∧ 𝜓)} | |
2 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
3 | inidm 4153 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → (𝐴 ∩ 𝐴) = 𝐴) |
5 | 2, 4 | bj-rabeqd 35093 | . . 3 ⊢ (⊤ → {𝑥 ∈ (𝐴 ∩ 𝐴) ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}) |
6 | 5 | mptru 1546 | . 2 ⊢ {𝑥 ∈ (𝐴 ∩ 𝐴) ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
7 | 1, 6 | eqtri 2766 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ⊤wtru 1540 {crab 3068 ∩ cin 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3432 df-in 3894 |
This theorem is referenced by: (None) |
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