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Mirrors > Home > MPE Home > Th. List > eqsn | Structured version Visualization version Unicode version |
Description: Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
eqsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2795 | . . 3 | |
2 | biorf 420 | . . 3 | |
3 | 1, 2 | sylbi 207 | . 2 |
4 | dfss3 3592 | . . 3 | |
5 | sssn 4358 | . . 3 | |
6 | velsn 4193 | . . . 4 | |
7 | 6 | ralbii 2980 | . . 3 |
8 | 4, 5, 7 | 3bitr3i 290 | . 2 |
9 | 3, 8 | syl6bb 276 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wceq 1483 wcel 1990 wne 2794 wral 2912 wss 3574 c0 3915 csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 |
This theorem is referenced by: issn 4363 zornn0g 9327 hashgt12el 13210 hashgt12el2 13211 hashge2el2dif 13262 lssne0 18951 qtopeu 21519 rngoueqz 33739 mapdm0OLD 39383 lmod0rng 41868 |
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