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Mathbox for Peter Mazsa |
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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 38104 | . 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴) | |
2 | relssr 38102 | . . 3 ⊢ Rel S | |
3 | 2 | brrelex1i 5734 | . 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V) |
4 | 1, 3 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 Vcvv 3461 class class class wbr 5149 S cssr 37782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-ssr 38100 |
This theorem is referenced by: (None) |
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