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Mirrors > Home > MPE Home > Th. List > noelOLD | Structured version Visualization version GIF version |
Description: Obsolete version of noel 4295 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 4102 | . . 3 ⊢ (𝐴 ∈ (V ∖ V) → 𝐴 ∈ V) | |
2 | eldifn 4103 | . . 3 ⊢ (𝐴 ∈ (V ∖ V) → ¬ 𝐴 ∈ V) | |
3 | 1, 2 | pm2.65i 196 | . 2 ⊢ ¬ 𝐴 ∈ (V ∖ V) |
4 | df-nul 4291 | . . 3 ⊢ ∅ = (V ∖ V) | |
5 | 4 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ ∅ ↔ 𝐴 ∈ (V ∖ V)) |
6 | 3, 5 | mtbir 325 | 1 ⊢ ¬ 𝐴 ∈ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∅c0 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-nul 4291 |
This theorem is referenced by: (None) |
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