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Mirrors > Home > MPE Home > Th. List > Mathboxes > distel | Structured version Visualization version GIF version |
Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5233 and elirrv 9126.) (Contributed by Scott Fenton, 15-Dec-2010.) |
Ref | Expression |
---|---|
distel | ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el 5233 | . . 3 ⊢ ∃𝑧 𝑥 ∈ 𝑧 | |
2 | df-ex 1787 | . . . 4 ⊢ (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧) | |
3 | nfnae 2433 | . . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥 | |
4 | dveel1 2460 | . . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧)) | |
5 | 3, 4 | nf5d 2287 | . . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝑧) |
6 | 5 | nfnd 1864 | . . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥 ∈ 𝑧) |
7 | elequ2 2128 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦)) | |
8 | 7 | notbid 321 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑦)) |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑦))) |
10 | 3, 6, 9 | cbvald 2406 | . . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ∀𝑦 ¬ 𝑥 ∈ 𝑦)) |
11 | 10 | notbid 321 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)) |
12 | 2, 11 | syl5bb 286 | . . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)) |
13 | 1, 12 | mpbii 236 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦) |
14 | elirrv 9126 | . . . . 5 ⊢ ¬ 𝑦 ∈ 𝑦 | |
15 | elequ1 2120 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
16 | 14, 15 | mtbii 329 | . . . 4 ⊢ (𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝑦) |
17 | 16 | alimi 1818 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥 ∈ 𝑦) |
18 | 17 | con3i 157 | . 2 ⊢ (¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥) |
19 | 13, 18 | impbii 212 | 1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-13 2371 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-reg 9122 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-v 3399 df-dif 3844 df-un 3846 df-nul 4210 df-sn 4514 df-pr 4516 |
This theorem is referenced by: (None) |
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