Theorem nninfinfwlpo 7510

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 7463). (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 6934	. . . . . 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 5699	. . . . . . . . 9 |- ( q = z -> ( ( f ` q ) = (/) <-> ( f ` z ) = (/) ) )
4	3	cbvrexv 2787	. . . . . . . 8 |- ( E. q e. suc j ( f ` q ) = (/) <-> E. z e. suc j ( f ` z ) = (/) )
5		suceq 4542	. . . . . . . . 9 |- ( j = k -> suc j = suc k )
6	5	rexeqdv 2756	. . . . . . . 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 3659	. . . . . 6 |- ( j = k -> if ( E. q e. suc j ( f ` q ) = (/) , (/) , 1o ) = if ( E. z e. suc k ( f ` z ) = (/) , (/) , 1o ) )
9	8	cbvmptv 4222	. . . . 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 2239	. . . . . . . . . . 11 |- ( i = k -> 1o = 1o )
13	12	cbvmptv 4222	. . . . . . . . . 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 2253	. . . . . . . 8 |- ( x = z -> ( x = ( i e. om |-> 1o ) <-> z = ( k e. om |-> 1o ) ) )
16	15	dcbid 850	. . . . . . 7 |- ( x = z -> (DECID x = ( i e. om |-> 1o ) <-> DECID z = ( k e. om |-> 1o ) ) )
17	16	cbvralv 2786	. . . . . 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 7508	. . . 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 2623	. . 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 4735	. . . 4 |- om e. _V
22		iswomnimap 7496	. . . 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 7501	. . 3 |- ( ( om e. WOmni /\ x e. ℕ∞ ) -> DECID x = ( i e. om |-> 1o ) )
28	27	ralrimiva 2623	. 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 846   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   (/)c0 3520   ifcif 3635   |-> cmpt 4187   suc csuc 4505   omcom 4732   -->wf 5368   ` cfv 5372  (class class class)co 6075   1oc1o 6670   2oc2o 6671   ^m cmap 6912  ℕ_∞xnninf 7449  WOmnicwomni 7493

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1o 6677  df-2o 6678  df-er 6797  df-map 6914  df-en 7013  df-fin 7015  df-nninf 7450  df-womni 7494

This theorem is referenced by: (None)