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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 4494). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4515). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Ref | Expression |
---|---|
nnsucuniel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3418 | . . . . . . 7 | |
2 | uni0 3823 | . . . . . . . 8 | |
3 | 2 | eleq2i 2237 | . . . . . . 7 |
4 | 1, 3 | mtbir 666 | . . . . . 6 |
5 | unieq 3805 | . . . . . . 7 | |
6 | 5 | eleq2d 2240 | . . . . . 6 |
7 | 4, 6 | mtbiri 670 | . . . . 5 |
8 | 7 | pm2.21d 614 | . . . 4 |
9 | 8 | adantl 275 | . . 3 |
10 | unieq 3805 | . . . . . . . . . . . 12 | |
11 | 10 | eleq2d 2240 | . . . . . . . . . . 11 |
12 | 11 | ad2antll 488 | . . . . . . . . . 10 |
13 | 12 | biimpa 294 | . . . . . . . . 9 |
14 | simplrl 530 | . . . . . . . . . . 11 | |
15 | nnord 4596 | . . . . . . . . . . . . 13 | |
16 | ordtr 4363 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . 12 |
18 | vex 2733 | . . . . . . . . . . . . 13 | |
19 | 18 | unisuc 4398 | . . . . . . . . . . . 12 |
20 | 17, 19 | sylib 121 | . . . . . . . . . . 11 |
21 | 14, 20 | syl 14 | . . . . . . . . . 10 |
22 | 21 | eleq2d 2240 | . . . . . . . . 9 |
23 | 13, 22 | mpbid 146 | . . . . . . . 8 |
24 | nnsucelsuc 6470 | . . . . . . . . 9 | |
25 | 14, 24 | syl 14 | . . . . . . . 8 |
26 | 23, 25 | mpbid 146 | . . . . . . 7 |
27 | simplrr 531 | . . . . . . 7 | |
28 | 26, 27 | eleqtrrd 2250 | . . . . . 6 |
29 | 28 | ex 114 | . . . . 5 |
30 | 29 | rexlimdvaa 2588 | . . . 4 |
31 | 30 | imp 123 | . . 3 |
32 | nn0suc 4588 | . . 3 | |
33 | 9, 31, 32 | mpjaodan 793 | . 2 |
34 | sucunielr 4494 | . 2 | |
35 | 33, 34 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 c0 3414 cuni 3796 wtr 4087 word 4347 csuc 4350 com 4574 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 |
This theorem is referenced by: (None) |
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