Theorem nninfinfwlpo 7343

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 7296). (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 6815	. . . . . 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 5635	. . . . . . . . 9 |- ( q = z -> ( ( f ` q ) = (/) <-> ( f ` z ) = (/) ) )
4	3	cbvrexv 2766	. . . . . . . 8 |- ( E. q e. suc j ( f ` q ) = (/) <-> E. z e. suc j ( f ` z ) = (/) )
5		suceq 4492	. . . . . . . . 9 |- ( j = k -> suc j = suc k )
6	5	rexeqdv 2735	. . . . . . . 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 3624	. . . . . 6 |- ( j = k -> if ( E. q e. suc j ( f ` q ) = (/) , (/) , 1o ) = if ( E. z e. suc k ( f ` z ) = (/) , (/) , 1o ) )
9	8	cbvmptv 4179	. . . . 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 2230	. . . . . . . . . . 11 |- ( i = k -> 1o = 1o )
13	12	cbvmptv 4179	. . . . . . . . . 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 2244	. . . . . . . 8 |- ( x = z -> ( x = ( i e. om |-> 1o ) <-> z = ( k e. om |-> 1o ) ) )
16	15	dcbid 843	. . . . . . 7 |- ( x = z -> (DECID x = ( i e. om |-> 1o ) <-> DECID z = ( k e. om |-> 1o ) ) )
17	16	cbvralv 2765	. . . . . 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 7341	. . . 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 2603	. . 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 4684	. . . 4 |- om e. _V
22		iswomnimap 7329	. . . 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 7334	. . 3 |- ( ( om e. WOmni /\ x e. ℕ∞ ) -> DECID x = ( i e. om |-> 1o ) )
28	27	ralrimiva 2603	. 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 839   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2799   (/)c0 3491   ifcif 3602   |-> cmpt 4144   suc csuc 4455   omcom 4681   -->wf 5313   ` cfv 5317  (class class class)co 6000   1oc1o 6553   2oc2o 6554   ^m cmap 6793  ℕ_∞xnninf 7282  WOmnicwomni 7326

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1o 6560  df-2o 6561  df-er 6678  df-map 6795  df-en 6886  df-fin 6888  df-nninf 7283  df-womni 7327

This theorem is referenced by: (None)