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Mirrors > Home > MPE Home > Th. List > sorpssuni | Structured version Visualization version Unicode version |
Description: In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
sorpssuni | [] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sorpssi 6943 | . . . . . . . . . 10 [] | |
2 | 1 | anassrs 680 | . . . . . . . . 9 [] |
3 | sspss 3706 | . . . . . . . . . . 11 | |
4 | orel1 397 | . . . . . . . . . . . 12 | |
5 | eqimss2 3658 | . . . . . . . . . . . 12 | |
6 | 4, 5 | syl6com 37 | . . . . . . . . . . 11 |
7 | 3, 6 | sylbi 207 | . . . . . . . . . 10 |
8 | ax-1 6 | . . . . . . . . . 10 | |
9 | 7, 8 | jaoi 394 | . . . . . . . . 9 |
10 | 2, 9 | syl 17 | . . . . . . . 8 [] |
11 | 10 | ralimdva 2962 | . . . . . . 7 [] |
12 | 11 | 3impia 1261 | . . . . . 6 [] |
13 | unissb 4469 | . . . . . 6 | |
14 | 12, 13 | sylibr 224 | . . . . 5 [] |
15 | elssuni 4467 | . . . . . 6 | |
16 | 15 | 3ad2ant2 1083 | . . . . 5 [] |
17 | 14, 16 | eqssd 3620 | . . . 4 [] |
18 | simp2 1062 | . . . 4 [] | |
19 | 17, 18 | eqeltrd 2701 | . . 3 [] |
20 | 19 | rexlimdv3a 3033 | . 2 [] |
21 | elssuni 4467 | . . . . 5 | |
22 | ssnpss 3710 | . . . . 5 | |
23 | 21, 22 | syl 17 | . . . 4 |
24 | 23 | rgen 2922 | . . 3 |
25 | psseq1 3694 | . . . . . 6 | |
26 | 25 | notbid 308 | . . . . 5 |
27 | 26 | ralbidv 2986 | . . . 4 |
28 | 27 | rspcev 3309 | . . 3 |
29 | 24, 28 | mpan2 707 | . 2 |
30 | 20, 29 | impbid1 215 | 1 [] |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wss 3574 wpss 3575 cuni 4436 wor 5034 [] crpss 6936 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-so 5036 df-xp 5120 df-rel 5121 df-rpss 6937 |
This theorem is referenced by: fin2i2 9140 isfin2-2 9141 fin12 9235 |
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