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Mirrors > Home > MPE Home > Th. List > tpss | Structured version Visualization version Unicode version |
Description: A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
tpss.1 | |
tpss.2 | |
tpss.3 |
Ref | Expression |
---|---|
tpss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss 3787 | . 2 | |
2 | df-3an 1039 | . . 3 | |
3 | tpss.1 | . . . . 5 | |
4 | tpss.2 | . . . . 5 | |
5 | 3, 4 | prss 4351 | . . . 4 |
6 | tpss.3 | . . . . 5 | |
7 | 6 | snss 4316 | . . . 4 |
8 | 5, 7 | anbi12i 733 | . . 3 |
9 | 2, 8 | bitri 264 | . 2 |
10 | df-tp 4182 | . . 3 | |
11 | 10 | sseq1i 3629 | . 2 |
12 | 1, 9, 11 | 3bitr4i 292 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 w3a 1037 wcel 1990 cvv 3200 cun 3572 wss 3574 csn 4177 cpr 4179 ctp 4181 |
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-v 3202 df-un 3579 df-in 3581 df-ss 3588 df-sn 4178 df-pr 4180 df-tp 4182 |
This theorem is referenced by: 1cubr 24569 konigsberglem4 27117 rabren3dioph 37379 fourierdlem102 40425 fourierdlem114 40437 nnsum4primesodd 41684 nnsum4primesoddALTV 41685 |
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