Theorem onsucelsucexmidlem 4562

Description: Lemma for onsucelsucexmid 4563. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5910), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4552. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
onsucelsucexmidlem	|- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On

Distinct variable group:   ph, x

Proof of Theorem onsucelsucexmidlem

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. z )
2		noel 3451	. . . . . . . . . 10 |- -. y e. (/)
3		eleq2 2257	. . . . . . . . . 10 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
4	2, 3	mtbiri 676	. . . . . . . . 9 |- ( z = (/) -> -. y e. z )
5	4	adantl 277	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> -. y e. z )
6	1, 5	pm2.21dd 621	. . . . . . 7 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
7	6	ex 115	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = (/) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
8		eleq2 2257	. . . . . . . . . . 11 |- ( z = { (/) } -> ( y e. z <-> y e. { (/) } ) )
9	8	biimpac 298	. . . . . . . . . 10 |- ( ( y e. z /\ z = { (/) } ) -> y e. { (/) } )
10		velsn 3636	. . . . . . . . . 10 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	sylib 122	. . . . . . . . 9 |- ( ( y e. z /\ z = { (/) } ) -> y = (/) )
12		onsucelsucexmidlem1 4561	. . . . . . . . 9 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
13	11, 12	eqeltrdi 2284	. . . . . . . 8 |- ( ( y e. z /\ z = { (/) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
14	13	ex 115	. . . . . . 7 |- ( y e. z -> ( z = { (/) } -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
15	14	adantr 276	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = { (/) } -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
16		elrabi 2914	. . . . . . . 8 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> z e. { (/) , { (/) } } )
17		vex 2763	. . . . . . . . 9 |- z e. _V
18	17	elpr 3640	. . . . . . . 8 |- ( z e. { (/) , { (/) } } <-> ( z = (/) \/ z = { (/) } ) )
19	16, 18	sylib 122	. . . . . . 7 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> ( z = (/) \/ z = { (/) } ) )
20	19	adantl 277	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = (/) \/ z = { (/) } ) )
21	7, 15, 20	mpjaod 719	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
22	21	gen2 1461	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
23		dftr2 4130	. . . 4 |- ( Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> A. y A. z ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
24	22, 23	mpbir 146	. . 3 |- Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
25		ssrab2 3265	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } }
26		2ordpr 4557	. . 3 |- Ord { (/) , { (/) } }
27		trssord 4412	. . 3 |- ( ( Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } /\ { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } } /\ Ord { (/) , { (/) } } ) -> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
28	24, 25, 26, 27	mp3an 1348	. 2 |- Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
29		pp0ex 4219	. . . 4 |- { (/) , { (/) } } e. _V
30	29	rabex 4174	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. _V
31	30	elon 4406	. 2 |- ( { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On <-> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
32	28, 31	mpbir 146	1 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   A.wal 1362   = wceq 1364   e. wcel 2164   {crab 2476   C_ wss 3154   (/)c0 3447   {csn 3619   {cpr 3620   Tr wtr 4128   Ord word 4394   Oncon0 4395

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403

This theorem is referenced by:  onsucelsucexmid  4563  acexmidlemcase  5914  acexmidlemv  5917