|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 3233 | . 2 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∅) | |
| 2 | eq0 4350 | . . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
| 3 | 2 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) | 
| 4 | 1, 3 | bitri 275 | 1 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: n0el 4364 axinf2 9680 zfinf2 9682 gneispace 44147 | 
| Copyright terms: Public domain | W3C validator |