Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 0elnn | Unicode version |
Description: A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Ref | Expression |
---|---|
0elnn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2171 | . . 3 | |
2 | eleq2 2228 | . . 3 | |
3 | 1, 2 | orbi12d 783 | . 2 |
4 | eqeq1 2171 | . . 3 | |
5 | eleq2 2228 | . . 3 | |
6 | 4, 5 | orbi12d 783 | . 2 |
7 | eqeq1 2171 | . . 3 | |
8 | eleq2 2228 | . . 3 | |
9 | 7, 8 | orbi12d 783 | . 2 |
10 | eqeq1 2171 | . . 3 | |
11 | eleq2 2228 | . . 3 | |
12 | 10, 11 | orbi12d 783 | . 2 |
13 | eqid 2164 | . . 3 | |
14 | 13 | orci 721 | . 2 |
15 | 0ex 4103 | . . . . . . 7 | |
16 | 15 | sucid 4389 | . . . . . 6 |
17 | suceq 4374 | . . . . . 6 | |
18 | 16, 17 | eleqtrrid 2254 | . . . . 5 |
19 | 18 | a1i 9 | . . . 4 |
20 | sssucid 4387 | . . . . . 6 | |
21 | 20 | a1i 9 | . . . . 5 |
22 | 21 | sseld 3136 | . . . 4 |
23 | 19, 22 | jaod 707 | . . 3 |
24 | olc 701 | . . 3 | |
25 | 23, 24 | syl6 33 | . 2 |
26 | 3, 6, 9, 12, 14, 25 | finds 4571 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 698 wceq 1342 wcel 2135 wss 3111 c0 3404 csuc 4337 com 4561 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 |
This theorem is referenced by: nn0eln0 4591 nnsucsssuc 6451 nntri3or 6452 nnm00 6488 ssfilem 6832 diffitest 6844 fiintim 6885 enumct 7071 nnnninfeq 7083 elni2 7246 enq0tr 7366 bj-charfunr 13533 |
Copyright terms: Public domain | W3C validator |