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Mathbox for Alexander van der Vekens |
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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 9537 | . . 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 2107 ∀wral 3061 Vcvv 3444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-pr 5385 ax-reg 9533 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-v 3446 df-un 3916 df-sn 4588 df-pr 4590 |
This theorem is referenced by: (None) |
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