Theorem onsucelsucexmidlem 4513

Description: Lemma for onsucelsucexmid 4514. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5844), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4503. (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 524	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. z )
2		noel 3418	. . . . . . . . . 10 |- -. y e. (/)
3		eleq2 2234	. . . . . . . . . 10 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
4	2, 3	mtbiri 670	. . . . . . . . 9 |- ( z = (/) -> -. y e. z )
5	4	adantl 275	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> -. y e. z )
6	1, 5	pm2.21dd 615	. . . . . . 7 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
7	6	ex 114	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = (/) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
8		eleq2 2234	. . . . . . . . . . 11 |- ( z = { (/) } -> ( y e. z <-> y e. { (/) } ) )
9	8	biimpac 296	. . . . . . . . . 10 |- ( ( y e. z /\ z = { (/) } ) -> y e. { (/) } )
10		velsn 3600	. . . . . . . . . 10 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	sylib 121	. . . . . . . . 9 |- ( ( y e. z /\ z = { (/) } ) -> y = (/) )
12		onsucelsucexmidlem1 4512	. . . . . . . . 9 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
13	11, 12	eqeltrdi 2261	. . . . . . . 8 |- ( ( y e. z /\ z = { (/) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
14	13	ex 114	. . . . . . 7 |- ( y e. z -> ( z = { (/) } -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
15	14	adantr 274	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = { (/) } -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
16		elrabi 2883	. . . . . . . 8 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> z e. { (/) , { (/) } } )
17		vex 2733	. . . . . . . . 9 |- z e. _V
18	17	elpr 3604	. . . . . . . 8 |- ( z e. { (/) , { (/) } } <-> ( z = (/) \/ z = { (/) } ) )
19	16, 18	sylib 121	. . . . . . 7 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> ( z = (/) \/ z = { (/) } ) )
20	19	adantl 275	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = (/) \/ z = { (/) } ) )
21	7, 15, 20	mpjaod 713	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
22	21	gen2 1443	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
23		dftr2 4089	. . . 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 145	. . 3 |- Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
25		ssrab2 3232	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } }
26		2ordpr 4508	. . 3 |- Ord { (/) , { (/) } }
27		trssord 4365	. . 3 |- ( ( Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } /\ { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } } /\ Ord { (/) , { (/) } } ) -> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
28	24, 25, 26, 27	mp3an 1332	. 2 |- Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
29		pp0ex 4175	. . . 4 |- { (/) , { (/) } } e. _V
30	29	rabex 4133	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. _V
31	30	elon 4359	. 2 |- ( { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On <-> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
32	28, 31	mpbir 145	1 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 703   A.wal 1346   = wceq 1348   e. wcel 2141   {crab 2452   C_ wss 3121   (/)c0 3414   {csn 3583   {cpr 3584   Tr wtr 4087   Ord word 4347   Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by:  onsucelsucexmid  4514  acexmidlemcase  5848  acexmidlemv  5851