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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 3507 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | df-v 3497 | . . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
3 | 2 | eqeq2i 2834 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) |
4 | equid 2010 | . . . . . . 7 ⊢ 𝑥 = 𝑥 | |
5 | 4 | tbt 371 | . . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
6 | 5 | albii 1811 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
7 | alnex 1773 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
8 | abbi 2888 | . . . . 5 ⊢ (∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) | |
9 | 6, 7, 8 | 3bitr3ri 303 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
10 | 3, 9 | bitri 276 | . . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
11 | 10 | necon2abii 3066 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
12 | 1, 11 | bitri 276 | 1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2799 ≠ wne 3016 Vcvv 3495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-ne 3017 df-v 3497 |
This theorem is referenced by: (None) |
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