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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 3892 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)) | |
2 | n0rex 4166 | . 2 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴) | |
3 | 1, 2 | impel 501 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 ∃wrex 3118 ⊆ wss 3798 ∅c0 4146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-rex 3123 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 df-nul 4147 |
This theorem is referenced by: uhgrvd00 26839 |
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