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Mirrors > Home > MPE Home > Th. List > Mathboxes > pssexg | Structured version Visualization version GIF version |
Description: The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.) |
Ref | Expression |
---|---|
pssexg | ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssss 4107 | . 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssexg 5328 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | |
3 | 1, 2 | sylan 580 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 ⊊ wpss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-in 3969 df-ss 3979 df-pss 3982 |
This theorem is referenced by: (None) |
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