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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 9053 | . . 3 ⊢ ¬ 𝑥 ∈ 𝑥 | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (𝑥 ∈ 𝑥 → V ∈ 𝑥) |
3 | 2 | rgen 3147 | 1 ⊢ ∀𝑥 ∈ 𝑥 V ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 ∀wral 3137 Vcvv 3491 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-reg 9049 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-v 3493 df-dif 3932 df-un 3934 df-nul 4285 df-sn 4561 df-pr 4563 |
This theorem is referenced by: (None) |
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