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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1292 | 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 |
---|---|
bnj1292.1 | ⊢ 𝐴 = (𝐵 ∩ 𝐶) |
Ref | Expression |
---|---|
bnj1292 | ⊢ 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1292.1 | . 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
2 | inss1 4203 | . 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵 | |
3 | 1, 2 | eqsstri 3999 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∩ cin 3933 ⊆ wss 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-v 3495 df-in 3941 df-ss 3950 |
This theorem is referenced by: bnj1253 32282 bnj1286 32284 bnj1280 32285 bnj1296 32286 |
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