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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 35757 | . 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴) | |
2 | relssr 35755 | . . 3 ⊢ Rel S | |
3 | 2 | brrelex1i 5608 | . 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V) |
4 | 1, 3 | impbii 211 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 S cssr 35471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-ssr 35753 |
This theorem is referenced by: (None) |
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