Theorem nn0eln0 4541

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 4540	. 2 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
2		noel 3372	. . . . 5 |- -. (/) e. (/)
3		eleq2 2204	. . . . 5 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
4	2, 3	mtbiri 665	. . . 4 |- ( A = (/) -> -. (/) e. A )
5		nner 2313	. . . 4 |- ( A = (/) -> -. A =/= (/) )
6	4, 5	2falsed 692	. . 3 |- ( A = (/) -> ( (/) e. A <-> A =/= (/) ) )
7		id 19	. . . 4 |- ( (/) e. A -> (/) e. A )
8		ne0i 3374	. . . 4 |- ( (/) e. A -> A =/= (/) )
9	7, 8	2thd 174	. . 3 |- ( (/) e. A -> ( (/) e. A <-> A =/= (/) ) )
10	6, 9	jaoi 706	. 2 |- ( ( A = (/) \/ (/) e. A ) -> ( (/) e. A <-> A =/= (/) ) )
11	1, 10	syl 14	1 |- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   =/= wne 2309   (/)c0 3368   omcom 4512

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-iom 4513

This theorem is referenced by:  nnmord  6421  nnnninf  7031