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Mirrors > Home > MPE Home > Th. List > ssn0rex | Structured version Visualization version GIF version |
Description: There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
ssn0rex | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexv 4012 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)) | |
2 | n0rex 4315 | . 2 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴) | |
3 | 1, 2 | impel 507 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3911 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-rex 3071 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 df-nul 4284 |
This theorem is referenced by: uhgrvd00 28524 |
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