| 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 38964 | . 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴) | |
| 2 | relssr 38962 | . . 3 ⊢ Rel S | |
| 3 | 2 | brrelex1i 5677 | . 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 2121 Vcvv 3433 class class class wbr 5075 S cssr 38568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-ext 2713 ax-sep 5221 ax-pr 5365 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-sb 2075 df-clab 2720 df-cleq 2733 df-clel 2816 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-br 5076 df-opab 5138 df-xp 5627 df-rel 5628 df-ssr 38960 |
| This theorem is referenced by: (None) |
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