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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 3460 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | df-v 3449 | . . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
3 | 2 | eqeq2i 2746 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) |
4 | abbib 2805 | . . . . 5 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) | |
5 | equid 2016 | . . . . . . 7 ⊢ 𝑥 = 𝑥 | |
6 | 5 | tbt 370 | . . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
7 | 6 | albii 1822 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
8 | alnex 1784 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
9 | 4, 7, 8 | 3bitr2i 299 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
10 | 3, 9 | bitri 275 | . . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
11 | 10 | necon2abii 2991 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
12 | 1, 11 | bitri 275 | 1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ≠ wne 2940 Vcvv 3447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3449 |
This theorem is referenced by: (None) |
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