Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
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 4054 | . 2 ⊢ 𝐵 ⊆ 𝐴 |
3 | bnj1212.2 | . . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏)) | |
4 | 3 | simp2bi 1138 | . 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵) |
5 | 2, 4 | bnj1213 31969 | 1 ⊢ (𝜃 → 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 |
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-3an 1081 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-rab 3144 df-in 3940 df-ss 3949 |
This theorem is referenced by: bnj1204 32181 bnj1296 32190 bnj1415 32207 bnj1421 32211 bnj1442 32218 bnj1452 32221 bnj1489 32225 |
Copyright terms: Public domain | W3C validator |