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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 1834 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | 
| 4 | n0 4352 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 5 | df-rex 3070 | . 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 1778 ∈ wcel 2107 ≠ wne 2939 ∃wrex 3069 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-ne 2940 df-rex 3070 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: ssn0rex 4357 aks5lem7 42202 | 
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