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Mirrors > Home > ILE Home > Th. List > onsucelsucexmidlem | Unicode version |
Description: Lemma for onsucelsucexmid 4445. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5765), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4435. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
onsucelsucexmidlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . . . . . . 8 | |
2 | noel 3367 | . . . . . . . . . 10 | |
3 | eleq2 2203 | . . . . . . . . . 10 | |
4 | 2, 3 | mtbiri 664 | . . . . . . . . 9 |
5 | 4 | adantl 275 | . . . . . . . 8 |
6 | 1, 5 | pm2.21dd 609 | . . . . . . 7 |
7 | 6 | ex 114 | . . . . . 6 |
8 | eleq2 2203 | . . . . . . . . . . 11 | |
9 | 8 | biimpac 296 | . . . . . . . . . 10 |
10 | velsn 3544 | . . . . . . . . . 10 | |
11 | 9, 10 | sylib 121 | . . . . . . . . 9 |
12 | onsucelsucexmidlem1 4443 | . . . . . . . . 9 | |
13 | 11, 12 | eqeltrdi 2230 | . . . . . . . 8 |
14 | 13 | ex 114 | . . . . . . 7 |
15 | 14 | adantr 274 | . . . . . 6 |
16 | elrabi 2837 | . . . . . . . 8 | |
17 | vex 2689 | . . . . . . . . 9 | |
18 | 17 | elpr 3548 | . . . . . . . 8 |
19 | 16, 18 | sylib 121 | . . . . . . 7 |
20 | 19 | adantl 275 | . . . . . 6 |
21 | 7, 15, 20 | mpjaod 707 | . . . . 5 |
22 | 21 | gen2 1426 | . . . 4 |
23 | dftr2 4028 | . . . 4 | |
24 | 22, 23 | mpbir 145 | . . 3 |
25 | ssrab2 3182 | . . 3 | |
26 | 2ordpr 4439 | . . 3 | |
27 | trssord 4302 | . . 3 | |
28 | 24, 25, 26, 27 | mp3an 1315 | . 2 |
29 | pp0ex 4113 | . . . 4 | |
30 | 29 | rabex 4072 | . . 3 |
31 | 30 | elon 4296 | . 2 |
32 | 28, 31 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wal 1329 wceq 1331 wcel 1480 crab 2420 wss 3071 c0 3363 csn 3527 cpr 3528 wtr 4026 word 4284 con0 4285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: onsucelsucexmid 4445 acexmidlemcase 5769 acexmidlemv 5772 |
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