Theorem nn0eln0 4621

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 4620	. 2 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
2		noel 3428	. . . . 5 |- -. (/) e. (/)
3		eleq2 2241	. . . . 5 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
4	2, 3	mtbiri 675	. . . 4 |- ( A = (/) -> -. (/) e. A )
5		nner 2351	. . . 4 |- ( A = (/) -> -. A =/= (/) )
6	4, 5	2falsed 702	. . 3 |- ( A = (/) -> ( (/) e. A <-> A =/= (/) ) )
7		id 19	. . . 4 |- ( (/) e. A -> (/) e. A )
8		ne0i 3431	. . . 4 |- ( (/) e. A -> A =/= (/) )
9	7, 8	2thd 175	. . 3 |- ( (/) e. A -> ( (/) e. A <-> A =/= (/) ) )
10	6, 9	jaoi 716	. 2 |- ( ( A = (/) \/ (/) e. A ) -> ( (/) e. A <-> A =/= (/) ) )
11	1, 10	syl 14	1 |- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   =/= wne 2347   (/)c0 3424   omcom 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592

This theorem is referenced by:  nnmord  6520  bj-charfunr  14647