Theorem snsn0non 3120

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 3131). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 3236.

Assertion

Ref	Expression
snsn0non	|- -. {{(/)}} e. On

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		0ex 2706	. . . . 5 |- (/) e. V
2	1	snnz 2454	. . . 4 |- {(/)} =/= (/)
3	1	elsnc 2427	. . . . 5 |- ((/) e. {{(/)}} <-> (/) = {(/)})
4		eqcom 1474	. . . . 5 |- ((/) = {(/)} <-> {(/)} = (/))
5	3, 4	bitr 173	. . . 4 |- ((/) e. {{(/)}} <-> {(/)} = (/))
6	2, 5	nemtbir 1638	. . 3 |- -. (/) e. {{(/)}}
7	1	snid 2431	. . . 4 |- (/) e. {(/)}
8		ssel 2059	. . . 4 |- ({(/)} (_ {{(/)}} -> ((/) e. {(/)} -> (/) e. {{(/)}}))
9	7, 8	mpi 44	. . 3 |- ({(/)} (_ {{(/)}} -> (/) e. {{(/)}})
10	6, 9	mto 106	. 2 |- -. {(/)} (_ {{(/)}}
11		p0ex 2765	. . . 4 |- {(/)} e. V
12	11	snid 2431	. . 3 |- {(/)} e. {{(/)}}
13		onelsst 2995	. . 3 |- ({{(/)}} e. On -> ({(/)} e. {{(/)}} -> {(/)} (_ {{(/)}}))
14	12, 13	mpi 44	. 2 |- ({{(/)}} e. On -> {(/)} (_ {{(/)}})
15	10, 14	mto 106	1 |- -. {{(/)}} e. On

Syntax hints:  -. wn 2   = wceq 954   e. wcel 956   (_ wss 2043  (/)c0 2276  {csn 2405  Oncon0 2943

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-tr 2676  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947

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