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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ru | Structured version Visualization version GIF version |
Description: Remove dependency on ax-13 2354 (and df-v 3385) from Russell's paradox ru 3630 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4974 does require ax-8 2159 and ax-9 2166 since it requires df-clel 2793 and df-cleq 2790--- see bj-df-clel 33371 and bj-df-cleq 33376). Note the more economical use of bj-elissetv 33344 instead of isset 3393 to avoid use of df-v 3385. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ru | ⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ru1 33418 | . 2 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} | |
2 | bj-elissetv 33344 | . 2 ⊢ ({𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}) | |
3 | 1, 2 | mto 189 | 1 ⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1653 ∃wex 1875 ∈ wcel 2157 {cab 2783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 |
This theorem is referenced by: (None) |
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