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Mirrors > Home > ILE Home > Th. List > onsucelsucexmidlem | Unicode version |
Description: Lemma for onsucelsucexmid 4255. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5503), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4245. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
onsucelsucexmidlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 481 | . . . . . . . 8 | |
2 | noel 3228 | . . . . . . . . . 10 | |
3 | eleq2 2101 | . . . . . . . . . 10 | |
4 | 2, 3 | mtbiri 600 | . . . . . . . . 9 |
5 | 4 | adantl 262 | . . . . . . . 8 |
6 | 1, 5 | pm2.21dd 550 | . . . . . . 7 |
7 | 6 | ex 108 | . . . . . 6 |
8 | eleq2 2101 | . . . . . . . . . . 11 | |
9 | 8 | biimpac 282 | . . . . . . . . . 10 |
10 | velsn 3392 | . . . . . . . . . 10 | |
11 | 9, 10 | sylib 127 | . . . . . . . . 9 |
12 | onsucelsucexmidlem1 4253 | . . . . . . . . 9 | |
13 | 11, 12 | syl6eqel 2128 | . . . . . . . 8 |
14 | 13 | ex 108 | . . . . . . 7 |
15 | 14 | adantr 261 | . . . . . 6 |
16 | elrabi 2695 | . . . . . . . 8 | |
17 | vex 2560 | . . . . . . . . 9 | |
18 | 17 | elpr 3396 | . . . . . . . 8 |
19 | 16, 18 | sylib 127 | . . . . . . 7 |
20 | 19 | adantl 262 | . . . . . 6 |
21 | 7, 15, 20 | mpjaod 638 | . . . . 5 |
22 | 21 | gen2 1339 | . . . 4 |
23 | dftr2 3856 | . . . 4 | |
24 | 22, 23 | mpbir 134 | . . 3 |
25 | ssrab2 3025 | . . 3 | |
26 | 2ordpr 4249 | . . 3 | |
27 | trssord 4117 | . . 3 | |
28 | 24, 25, 26, 27 | mp3an 1232 | . 2 |
29 | pp0ex 3940 | . . . 4 | |
30 | 29 | rabex 3901 | . . 3 |
31 | 30 | elon 4111 | . 2 |
32 | 28, 31 | mpbir 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wcel 1393 crab 2310 wss 2917 c0 3224 csn 3375 cpr 3376 wtr 3854 word 4099 con0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: onsucelsucexmid 4255 acexmidlemcase 5507 acexmidlemv 5510 |
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