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Mirrors > Home > MPE Home > Th. List > snsspr2 | Structured version Visualization version GIF version |
Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.) |
Ref | Expression |
---|---|
snsspr2 | ⊢ {𝐵} ⊆ {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 4032 | . 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵}) | |
2 | df-pr 4438 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 1, 2 | sseqtr4i 3888 | 1 ⊢ {𝐵} ⊆ {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3821 ⊆ wss 3823 {csn 4435 {cpr 4437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-un 3828 df-in 3830 df-ss 3837 df-pr 4438 |
This theorem is referenced by: snsstp2 4618 ord3ex 5134 ltrelxr 10496 2strop 16454 2strop1 16457 phlip 16508 prdsco 16591 ipotset 17619 gsumpr 18822 lsppratlem4 19638 ex-res 27992 subfacp1lem2a 32012 dvh3dim3N 38030 algvsca 39178 corclrcl 39415 mnuprdlem4 39986 |
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