Theorem nninfinfwlpo 7297

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 7250). (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 6770	. . . . . 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 5598	. . . . . . . . 9 |- ( q = z -> ( ( f ` q ) = (/) <-> ( f ` z ) = (/) ) )
4	3	cbvrexv 2740	. . . . . . . 8 |- ( E. q e. suc j ( f ` q ) = (/) <-> E. z e. suc j ( f ` z ) = (/) )
5		suceq 4457	. . . . . . . . 9 |- ( j = k -> suc j = suc k )
6	5	rexeqdv 2710	. . . . . . . 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 3597	. . . . . 6 |- ( j = k -> if ( E. q e. suc j ( f ` q ) = (/) , (/) , 1o ) = if ( E. z e. suc k ( f ` z ) = (/) , (/) , 1o ) )
9	8	cbvmptv 4148	. . . . 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 2207	. . . . . . . . . . 11 |- ( i = k -> 1o = 1o )
13	12	cbvmptv 4148	. . . . . . . . . 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 2221	. . . . . . . 8 |- ( x = z -> ( x = ( i e. om |-> 1o ) <-> z = ( k e. om |-> 1o ) ) )
16	15	dcbid 840	. . . . . . 7 |- ( x = z -> (DECID x = ( i e. om |-> 1o ) <-> DECID z = ( k e. om |-> 1o ) ) )
17	16	cbvralv 2739	. . . . . 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 7295	. . . 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 2580	. . 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 4649	. . . 4 |- om e. _V
22		iswomnimap 7283	. . . 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 7288	. . 3 |- ( ( om e. WOmni /\ x e. ℕ∞ ) -> DECID x = ( i e. om |-> 1o ) )
28	27	ralrimiva 2580	. 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 836   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   _Vcvv 2773   (/)c0 3464   ifcif 3575   |-> cmpt 4113   suc csuc 4420   omcom 4646   -->wf 5276   ` cfv 5280  (class class class)co 5957   1oc1o 6508   2oc2o 6509   ^m cmap 6748  ℕ_∞xnninf 7236  WOmnicwomni 7280

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1o 6515  df-2o 6516  df-er 6633  df-map 6750  df-en 6841  df-fin 6843  df-nninf 7237  df-womni 7281

This theorem is referenced by: (None)