Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 1n0 | GIF version |
Description: Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) |
Ref | Expression |
---|---|
1n0 | ⊢ 1o ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6388 | . 2 ⊢ 1o = {∅} | |
2 | 0ex 4103 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | snnz 3689 | . 2 ⊢ {∅} ≠ ∅ |
4 | 1, 3 | eqnetri 2357 | 1 ⊢ 1o ≠ ∅ |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2334 ∅c0 3404 {csn 3570 1oc1o 6368 |
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-ext 2146 ax-nul 4102 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-v 2723 df-dif 3113 df-un 3115 df-nul 3405 df-sn 3576 df-suc 4343 df-1o 6375 |
This theorem is referenced by: xp01disj 6392 xp01disjl 6393 djulclb 7011 djuinr 7019 eldju2ndl 7028 djune 7034 updjudhf 7035 updjudhcoinrg 7037 nninfisollemne 7086 nninfisol 7088 exmidomni 7097 fodjum 7101 fodju0 7102 ismkvnex 7110 mkvprop 7113 omniwomnimkv 7122 1pi 7247 unct 12318 bj-charfunbi 13534 pwle2 13719 subctctexmid 13722 pw1nct 13724 peano3nninf 13728 nninfalllem1 13729 nninfall 13730 nninfsellemeq 13735 nninfsellemqall 13736 nninffeq 13741 |
Copyright terms: Public domain | W3C validator |