![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > issetssr | Structured version Visualization version GIF version |
Description: Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.) |
Ref | Expression |
---|---|
issetssr | ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brssrid 34880 | . 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴) | |
2 | relssr 34878 | . . 3 ⊢ Rel S | |
3 | brrelex1 5403 | . . 3 ⊢ ((Rel S ∧ 𝐴 S 𝐴) → 𝐴 ∈ V) | |
4 | 2, 3 | mpan 680 | . 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V) |
5 | 1, 4 | impbii 201 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2107 Vcvv 3398 class class class wbr 4886 Rel wrel 5360 S cssr 34609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-ssr 34876 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |