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Mirrors > Home > ILE Home > Th. List > nnsucuniel | Unicode version |
Description: Given an element of the union of a natural number , is an element of itself. The reverse direction holds for all ordinals (sucunielr 4421). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4441). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Ref | Expression |
---|---|
nnsucuniel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3362 | . . . . . . 7 | |
2 | uni0 3758 | . . . . . . . 8 | |
3 | 2 | eleq2i 2204 | . . . . . . 7 |
4 | 1, 3 | mtbir 660 | . . . . . 6 |
5 | unieq 3740 | . . . . . . 7 | |
6 | 5 | eleq2d 2207 | . . . . . 6 |
7 | 4, 6 | mtbiri 664 | . . . . 5 |
8 | 7 | pm2.21d 608 | . . . 4 |
9 | 8 | adantl 275 | . . 3 |
10 | unieq 3740 | . . . . . . . . . . . 12 | |
11 | 10 | eleq2d 2207 | . . . . . . . . . . 11 |
12 | 11 | ad2antll 482 | . . . . . . . . . 10 |
13 | 12 | biimpa 294 | . . . . . . . . 9 |
14 | simplrl 524 | . . . . . . . . . . 11 | |
15 | nnord 4520 | . . . . . . . . . . . . 13 | |
16 | ordtr 4295 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . 12 |
18 | vex 2684 | . . . . . . . . . . . . 13 | |
19 | 18 | unisuc 4330 | . . . . . . . . . . . 12 |
20 | 17, 19 | sylib 121 | . . . . . . . . . . 11 |
21 | 14, 20 | syl 14 | . . . . . . . . . 10 |
22 | 21 | eleq2d 2207 | . . . . . . . . 9 |
23 | 13, 22 | mpbid 146 | . . . . . . . 8 |
24 | nnsucelsuc 6380 | . . . . . . . . 9 | |
25 | 14, 24 | syl 14 | . . . . . . . 8 |
26 | 23, 25 | mpbid 146 | . . . . . . 7 |
27 | simplrr 525 | . . . . . . 7 | |
28 | 26, 27 | eleqtrrd 2217 | . . . . . 6 |
29 | 28 | ex 114 | . . . . 5 |
30 | 29 | rexlimdvaa 2548 | . . . 4 |
31 | 30 | imp 123 | . . 3 |
32 | nn0suc 4513 | . . 3 | |
33 | 9, 31, 32 | mpjaodan 787 | . 2 |
34 | sucunielr 4421 | . 2 | |
35 | 33, 34 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2415 c0 3358 cuni 3731 wtr 4021 word 4279 csuc 4282 com 4499 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 |
This theorem is referenced by: (None) |
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