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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 3916 | 1 ⊢ (𝜃 → 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ⊆ wss 3883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-v 3425 df-in 3890 df-ss 3900 |
This theorem is referenced by: bnj1212 32523 bnj1173 32726 bnj1296 32745 bnj1408 32760 bnj1452 32776 bnj1523 32795 |
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