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Mirrors > Home > MPE Home > Th. List > 0el | Structured version Visualization version GIF version |
Description: Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.) |
Ref | Expression |
---|---|
0el | ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3230 | . 2 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∅) | |
2 | eq0 4342 | . . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
3 | 2 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) |
4 | 1, 3 | bitri 274 | 1 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ∅c0 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rex 3071 df-dif 3950 df-nul 4322 |
This theorem is referenced by: n0el 4360 axinf2 9631 zfinf2 9633 gneispace 42870 |
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