Theorem onsucelsucexmidlem 4506

Description: Lemma for onsucelsucexmid 4507. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5833), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4496. (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 519	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. z )
2		noel 3413	. . . . . . . . . 10 |- -. y e. (/)
3		eleq2 2230	. . . . . . . . . 10 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
4	2, 3	mtbiri 665	. . . . . . . . 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 610	. . . . . . 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 2230	. . . . . . . . . . 11 |- ( z = { (/) } -> ( y e. z <-> y e. { (/) } ) )
9	8	biimpac 296	. . . . . . . . . 10 |- ( ( y e. z /\ z = { (/) } ) -> y e. { (/) } )
10		velsn 3593	. . . . . . . . . 10 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	sylib 121	. . . . . . . . 9 |- ( ( y e. z /\ z = { (/) } ) -> y = (/) )
12		onsucelsucexmidlem1 4505	. . . . . . . . 9 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
13	11, 12	eqeltrdi 2257	. . . . . . . 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 2879	. . . . . . . 8 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> z e. { (/) , { (/) } } )
17		vex 2729	. . . . . . . . 9 |- z e. _V
18	17	elpr 3597	. . . . . . . 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 708	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
22	21	gen2 1438	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
23		dftr2 4082	. . . 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 3227	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } }
26		2ordpr 4501	. . 3 |- Ord { (/) , { (/) } }
27		trssord 4358	. . 3 |- ( ( Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } /\ { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } } /\ Ord { (/) , { (/) } } ) -> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
28	24, 25, 26, 27	mp3an 1327	. 2 |- Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
29		pp0ex 4168	. . . 4 |- { (/) , { (/) } } e. _V
30	29	rabex 4126	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. _V
31	30	elon 4352	. 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 698   A.wal 1341   = wceq 1343   e. wcel 2136   {crab 2448   C_ wss 3116   (/)c0 3409   {csn 3576   {cpr 3577   Tr wtr 4080   Ord word 4340   Oncon0 4341

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  onsucelsucexmid  4507  acexmidlemcase  5837  acexmidlemv  5840