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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 4098 | . 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
| 2 | ssexg 5323 | . 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 2108 Vcvv 3480 ⊆ wss 3951 ⊊ wpss 3952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pss 3971 |
| This theorem is referenced by: (None) |
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