Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
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 3508 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | df-v 3498 | . . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
3 | 2 | eqeq2i 2836 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) |
4 | equid 2019 | . . . . . . 7 ⊢ 𝑥 = 𝑥 | |
5 | 4 | tbt 372 | . . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
6 | 5 | albii 1820 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥)) |
7 | alnex 1782 | . . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
8 | abbi 2890 | . . . . 5 ⊢ (∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥}) | |
9 | 6, 7, 8 | 3bitr3ri 304 | . . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
10 | 3, 9 | bitri 277 | . . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
11 | 10 | necon2abii 3068 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
12 | 1, 11 | bitri 277 | 1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ≠ wne 3018 Vcvv 3496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ne 3019 df-v 3498 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |