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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 548 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) |
3 | 2 | eximi 1829 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) |
4 | n0 4339 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
5 | df-rex 3063 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4i 292 | 1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 ∅c0 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-ne 2933 df-rex 3063 df-dif 3944 df-nul 4316 |
This theorem is referenced by: ssn0rex 4348 |
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