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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1212 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1212.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} |
| bnj1212.2 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏)) |
| Ref | Expression |
|---|---|
| bnj1212 | ⊢ (𝜃 → 𝑥 ∈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1212.1 | . . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
| 2 | 1 | ssrab3 4082 | . 2 ⊢ 𝐵 ⊆ 𝐴 |
| 3 | bnj1212.2 | . . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏)) | |
| 4 | 3 | simp2bi 1147 | . 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵) |
| 5 | 2, 4 | bnj1213 34812 | 1 ⊢ (𝜃 → 𝑥 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-ss 3968 |
| This theorem is referenced by: bnj1204 35026 bnj1296 35035 bnj1415 35052 bnj1421 35056 bnj1442 35063 bnj1452 35066 bnj1489 35070 |
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