| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ruOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of ru 3786 as of 20-Jun-2025. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2377. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ruOLD | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm5.19 386 | . . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦) | |
| 2 | eleq1w 2824 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦)) | |
| 3 | df-nel 3047 | . . . . . . . . 9 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥) | |
| 4 | id 22 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 5 | 4, 4 | eleq12d 2835 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) |
| 6 | 5 | notbid 318 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
| 7 | 3, 6 | bitrid 283 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
| 8 | 2, 7 | bibi12d 345 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))) |
| 9 | 8 | spvv 1996 | . . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)) |
| 10 | 1, 9 | mto 197 | . . . . 5 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) |
| 11 | eqabb 2881 | . . . . 5 ⊢ (𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥)) | |
| 12 | 10, 11 | mtbir 323 | . . . 4 ⊢ ¬ 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} |
| 13 | 12 | nex 1800 | . . 3 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} |
| 14 | isset 3494 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}) | |
| 15 | 13, 14 | mtbir 323 | . 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V |
| 16 | 15 | nelir 3049 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∉ wnel 3046 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nel 3047 df-v 3482 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |