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Mirrors > Home > MPE Home > Th. List > n0rex | Structured version Visualization version GIF version |
Description: There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
n0rex | ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) | |
2 | 1 | ancli 541 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) |
3 | 2 | eximi 1798 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) |
4 | n0 4191 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
5 | df-rex 3089 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4i 284 | 1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∃wex 1743 ∈ wcel 2051 ≠ wne 2962 ∃wrex 3084 ∅c0 4173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-11 2094 ax-12 2107 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-rex 3089 df-dif 3827 df-nul 4174 |
This theorem is referenced by: ssn0rex 4196 |
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