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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brssrid | Structured version Visualization version GIF version |
Description: Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.) |
Ref | Expression |
---|---|
brssrid | ⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4025 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | brssr 38405 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 S 𝐴 ↔ 𝐴 ⊆ 𝐴)) | |
3 | 1, 2 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 ⊆ wss 3970 class class class wbr 5169 S cssr 38086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-xp 5705 df-rel 5706 df-ssr 38402 |
This theorem is referenced by: issetssr 38407 |
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