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Mirrors > Home > MPE Home > Th. List > Mathboxes > elnev | Structured version Visualization version GIF version |
Description: Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.) |
Ref | Expression |
---|---|
elnev | ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isset 3180 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | df-v 3175 | . . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
3 | 2 | eqeq2i 2622 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) |
4 | equid 1926 | . . . . . . 7 ⊢ 𝑥 = 𝑥 | |
5 | 4 | tbt 358 | . . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
6 | 5 | albii 1737 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
7 | alnex 1697 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
8 | abbi 2724 | . . . . 5 ⊢ (∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) | |
9 | 6, 7, 8 | 3bitr3ri 290 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
10 | 3, 9 | bitri 263 | . . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
11 | 10 | necon2abii 2832 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
12 | 1, 11 | bitri 263 | 1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ne 2782 df-v 3175 |
This theorem is referenced by: (None) |
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