Theorem nn0eln0 4367

Description: A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)

Assertion

Ref	Expression
nn0eln0	|- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )

Proof of Theorem nn0eln0

Step	Hyp	Ref	Expression
1		0elnn 4366	. 2 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
2		noel 3262	. . . . 5 |- -. (/) e. (/)
3		eleq2 2143	. . . . 5 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
4	2, 3	mtbiri 633	. . . 4 |- ( A = (/) -> -. (/) e. A )
5		nner 2250	. . . 4 |- ( A = (/) -> -. A =/= (/) )
6	4, 5	2falsed 651	. . 3 |- ( A = (/) -> ( (/) e. A <-> A =/= (/) ) )
7		id 19	. . . 4 |- ( (/) e. A -> (/) e. A )
8		ne0i 3264	. . . 4 |- ( (/) e. A -> A =/= (/) )
9	7, 8	2thd 173	. . 3 |- ( (/) e. A -> ( (/) e. A <-> A =/= (/) ) )
10	6, 9	jaoi 669	. 2 |- ( ( A = (/) \/ (/) e. A ) -> ( (/) e. A <-> A =/= (/) ) )
11	1, 10	syl 14	1 |- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 662   = wceq 1285   e. wcel 1434   =/= wne 2246   (/)c0 3258   omcom 4339

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-iinf 4337

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340

This theorem is referenced by:  nnmord  6156