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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ralndv1 | Structured version Visualization version GIF version | ||
| Description: Example for a theorem about a restricted universal quantification in which the restricting class depends on (actually is) the bound variable: All sets containing themselves contain the universal class. (Contributed by AV, 24-Jun-2023.) |
| Ref | Expression |
|---|---|
| ralndv1 | ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elirrv 9636 | . . 3 ⊢ ¬ 𝑥 ∈ 𝑥 | |
| 2 | 1 | pm2.21i 119 | . 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥) |
| 3 | 2 | rgen 3063 | 1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∀wral 3061 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-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 |
| 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-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: (None) |
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