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Mirrors > Home > NFE Home > Th. List > evennnul | GIF version |
Description: An even number is nonempty. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
evennnul | ⊢ (A ∈ Evenfin → A ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2359 | . . . . . 6 ⊢ (x = A → (x = (n +c n) ↔ A = (n +c n))) | |
2 | 1 | rexbidv 2635 | . . . . 5 ⊢ (x = A → (∃n ∈ Nn x = (n +c n) ↔ ∃n ∈ Nn A = (n +c n))) |
3 | neeq1 2524 | . . . . 5 ⊢ (x = A → (x ≠ ∅ ↔ A ≠ ∅)) | |
4 | 2, 3 | anbi12d 691 | . . . 4 ⊢ (x = A → ((∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅) ↔ (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅))) |
5 | df-evenfin 4444 | . . . 4 ⊢ Evenfin = {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)} | |
6 | 4, 5 | elab2g 2987 | . . 3 ⊢ (A ∈ Evenfin → (A ∈ Evenfin ↔ (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅))) |
7 | 6 | ibi 232 | . 2 ⊢ (A ∈ Evenfin → (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅)) |
8 | 7 | simprd 449 | 1 ⊢ (A ∈ Evenfin → A ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 ∅c0 3550 Nn cnnc 4373 +c cplc 4375 Evenfin cevenfin 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-rex 2620 df-v 2861 df-evenfin 4444 |
This theorem is referenced by: evenoddnnnul 4514 vinf 4555 |
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