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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 3688 | . 2 ⊢ 𝐵 ⊆ 𝐴 |
3 | bnj1212.2 | . . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏)) | |
4 | 3 | simp2bi 1077 | . 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵) |
5 | 2, 4 | bnj1213 30869 | 1 ⊢ (𝜃 → 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-in 3581 df-ss 3588 |
This theorem is referenced by: bnj1204 31080 bnj1296 31089 bnj1415 31106 bnj1421 31110 bnj1442 31117 bnj1452 31120 bnj1489 31124 |
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