Theorem nninfinfwlpo 7422

Description: The point at infinity in ℕ_∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ_∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ_∞ corresponding to natural numbers are isolated (nninfisol 7375). (Contributed by Jim Kingdon, 25-Nov-2025.)

Assertion

Ref	Expression
nninfinfwlpo	|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )

Distinct variable group:   x, i

Proof of Theorem nninfinfwlpo

Dummy variables f k n z j q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 6882	. . . . . 6 |- ( f e. ( 2o ^m om ) -> f : om --> 2o )
2	1	adantl 277	. . . . 5 |- ( ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) /\ f e. ( 2o ^m om ) ) -> f : om --> 2o )
3		fveqeq2 5657	. . . . . . . . 9 |- ( q = z -> ( ( f ` q ) = (/) <-> ( f ` z ) = (/) ) )
4	3	cbvrexv 2769	. . . . . . . 8 |- ( E. q e. suc j ( f ` q ) = (/) <-> E. z e. suc j ( f ` z ) = (/) )
5		suceq 4505	. . . . . . . . 9 |- ( j = k -> suc j = suc k )
6	5	rexeqdv 2738	. . . . . . . 8 |- ( j = k -> ( E. z e. suc j ( f ` z ) = (/) <-> E. z e. suc k ( f ` z ) = (/) ) )
7	4, 6	bitrid 192	. . . . . . 7 |- ( j = k -> ( E. q e. suc j ( f ` q ) = (/) <-> E. z e. suc k ( f ` z ) = (/) ) )
8	7	ifbid 3631	. . . . . 6 |- ( j = k -> if ( E. q e. suc j ( f ` q ) = (/) , (/) , 1o ) = if ( E. z e. suc k ( f ` z ) = (/) , (/) , 1o ) )
9	8	cbvmptv 4190	. . . . 5 |- ( j e. om |-> if ( E. q e. suc j ( f ` q ) = (/) , (/) , 1o ) ) = ( k e. om |-> if ( E. z e. suc k ( f ` z ) = (/) , (/) , 1o ) )
10		simpl 109	. . . . . 6 |- ( ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) /\ f e. ( 2o ^m om ) ) -> A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) )
11		id 19	. . . . . . . . 9 |- ( x = z -> x = z )
12		eqidd 2232	. . . . . . . . . . 11 |- ( i = k -> 1o = 1o )
13	12	cbvmptv 4190	. . . . . . . . . 10 |- ( i e. om |-> 1o ) = ( k e. om |-> 1o )
14	13	a1i 9	. . . . . . . . 9 |- ( x = z -> ( i e. om |-> 1o ) = ( k e. om |-> 1o ) )
15	11, 14	eqeq12d 2246	. . . . . . . 8 |- ( x = z -> ( x = ( i e. om |-> 1o ) <-> z = ( k e. om |-> 1o ) ) )
16	15	dcbid 846	. . . . . . 7 |- ( x = z -> (DECID x = ( i e. om |-> 1o ) <-> DECID z = ( k e. om |-> 1o ) ) )
17	16	cbvralv 2768	. . . . . 6 |- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> A. z e. ℕ∞ DECID z = ( k e. om |-> 1o ) )
18	10, 17	sylib 122	. . . . 5 |- ( ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) /\ f e. ( 2o ^m om ) ) -> A. z e. ℕ∞ DECID z = ( k e. om |-> 1o ) )
19	2, 9, 18	nninfinfwlpolem 7420	. . . 4 |- ( ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) /\ f e. ( 2o ^m om ) ) -> DECID A. n e. om ( f ` n ) = 1o )
20	19	ralrimiva 2606	. . 3 |- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) -> A. f e. ( 2o ^m om )DECID A. n e. om ( f ` n ) = 1o )
21		omex 4697	. . . 4 |- om e. _V
22		iswomnimap 7408	. . . 4 |- ( om e. _V -> ( om e. WOmni <-> A. f e. ( 2o ^m om )DECID A. n e. om ( f ` n ) = 1o ) )
23	21, 22	ax-mp 5	. . 3 |- ( om e. WOmni <-> A. f e. ( 2o ^m om )DECID A. n e. om ( f ` n ) = 1o )
24	20, 23	sylibr 134	. 2 |- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) -> om e. WOmni )
25		simpl 109	. . . 4 |- ( ( om e. WOmni /\ x e. ℕ∞ ) -> om e. WOmni )
26		simpr 110	. . . 4 |- ( ( om e. WOmni /\ x e. ℕ∞ ) -> x e. ℕ∞ )
27	25, 26	nninfdcinf 7413	. . 3 |- ( ( om e. WOmni /\ x e. ℕ∞ ) -> DECID x = ( i e. om |-> 1o ) )
28	27	ralrimiva 2606	. 2 |- ( om e. WOmni -> A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) )
29	24, 28	impbii 126	1 |- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )

Syntax hints:   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   _Vcvv 2803   (/)c0 3496   ifcif 3607   |-> cmpt 4155   suc csuc 4468   omcom 4694   -->wf 5329   ` cfv 5333  (class class class)co 6028   1oc1o 6618   2oc2o 6619   ^m cmap 6860  ℕ_∞xnninf 7361  WOmnicwomni 7405

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-fin 6955  df-nninf 7362  df-womni 7406

This theorem is referenced by: (None)