![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > noelOLD | Structured version Visualization version GIF version |
Description: Obsolete version of noel 4183 as of 3-May-2023. The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
noelOLD | ⊢ ¬ 𝐴 ∈ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 3993 | . . 3 ⊢ (𝐴 ∈ (V ∖ V) → 𝐴 ∈ V) | |
2 | eldifn 3994 | . . 3 ⊢ (𝐴 ∈ (V ∖ V) → ¬ 𝐴 ∈ V) | |
3 | 1, 2 | pm2.65i 186 | . 2 ⊢ ¬ 𝐴 ∈ (V ∖ V) |
4 | df-nul 4179 | . . 3 ⊢ ∅ = (V ∖ V) | |
5 | 4 | eleq2i 2857 | . 2 ⊢ (𝐴 ∈ ∅ ↔ 𝐴 ∈ (V ∖ V)) |
6 | 3, 5 | mtbir 315 | 1 ⊢ ¬ 𝐴 ∈ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2050 Vcvv 3415 ∖ cdif 3826 ∅c0 4178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-v 3417 df-dif 3832 df-nul 4179 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |