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Mirrors > Home > NFE Home > Th. List > rabeq0 | GIF version |
Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) |
Ref | Expression |
---|---|
rabeq0 | ⊢ ({x ∈ A ∣ φ} = ∅ ↔ ∀x ∈ A ¬ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralnex 2624 | . 2 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ) | |
2 | rabn0 3570 | . . 3 ⊢ ({x ∈ A ∣ φ} ≠ ∅ ↔ ∃x ∈ A φ) | |
3 | 2 | necon1bbii 2568 | . 2 ⊢ (¬ ∃x ∈ A φ ↔ {x ∈ A ∣ φ} = ∅) |
4 | 1, 3 | bitr2i 241 | 1 ⊢ ({x ∈ A ∣ φ} = ∅ ↔ ∀x ∈ A ¬ φ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 = wceq 1642 ∀wral 2614 ∃wrex 2615 {crab 2618 ∅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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: rabnc 3574 nmembers1lem2 6269 |
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