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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1293 | 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 |
---|---|
bnj1293.1 | ⊢ 𝐴 = (𝐵 ∩ 𝐶) |
Ref | Expression |
---|---|
bnj1293 | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1293.1 | . 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
2 | inss2 4205 | . 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐶 | |
3 | 1, 2 | eqsstri 4000 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∩ cin 3934 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 |
This theorem is referenced by: bnj1253 32284 bnj1286 32286 bnj1280 32287 bnj1296 32288 |
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