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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 4094 | . 2 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssexg 5322 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | |
3 | 1, 2 | sylan 578 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2104 Vcvv 3472 ⊆ wss 3947 ⊊ wpss 3948 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-in 3954 df-ss 3964 df-pss 3966 |
This theorem is referenced by: (None) |
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