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Mirrors > Home > NFE Home > Th. List > rexn0 | GIF version |
Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
Ref | Expression |
---|---|
rexn0 | ⊢ (∃x ∈ A φ → A ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3556 | . . 3 ⊢ (x ∈ A → A ≠ ∅) | |
2 | 1 | a1d 22 | . 2 ⊢ (x ∈ A → (φ → A ≠ ∅)) |
3 | 2 | rexlimiv 2732 | 1 ⊢ (∃x ∈ A φ → A ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 ∅c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-nan 1288 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-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: (None) |
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