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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1213 | 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 |
---|---|
bnj1213.1 | ⊢ 𝐴 ⊆ 𝐵 |
bnj1213.2 | ⊢ (𝜃 → 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
bnj1213 | ⊢ (𝜃 → 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1213.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | bnj1213.2 | . 2 ⊢ (𝜃 → 𝑥 ∈ 𝐴) | |
3 | 1, 2 | sseldi 3967 | 1 ⊢ (𝜃 → 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-in 3945 df-ss 3954 |
This theorem is referenced by: bnj1212 32073 bnj1173 32276 bnj1296 32295 bnj1408 32310 bnj1452 32326 bnj1523 32345 |
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