Theorem onsucelsucexmidlem 4272

Description: Lemma for onsucelsucexmid 4273. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5523), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4263. (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 495	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. z )
2		noel 3255	. . . . . . . . . 10 |- -. y e. (/)
3		eleq2 2142	. . . . . . . . . 10 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
4	2, 3	mtbiri 632	. . . . . . . . 9 |- ( z = (/) -> -. y e. z )
5	4	adantl 271	. . . . . . . 8 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> -. y e. z )
6	1, 5	pm2.21dd 582	. . . . . . 7 |- ( ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) /\ z = (/) ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
7	6	ex 113	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = (/) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
8		eleq2 2142	. . . . . . . . . . 11 |- ( z = { (/) } -> ( y e. z <-> y e. { (/) } ) )
9	8	biimpac 292	. . . . . . . . . 10 |- ( ( y e. z /\ z = { (/) } ) -> y e. { (/) } )
10		velsn 3415	. . . . . . . . . 10 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	sylib 120	. . . . . . . . 9 |- ( ( y e. z /\ z = { (/) } ) -> y = (/) )
12		onsucelsucexmidlem1 4271	. . . . . . . . 9 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
13	11, 12	syl6eqel 2169	. . . . . . . 8 |- ( ( y e. z /\ z = { (/) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
14	13	ex 113	. . . . . . 7 |- ( y e. z -> ( z = { (/) } -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
15	14	adantr 270	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = { (/) } -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) )
16		elrabi 2746	. . . . . . . 8 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> z e. { (/) , { (/) } } )
17		vex 2604	. . . . . . . . 9 |- z e. _V
18	17	elpr 3419	. . . . . . . 8 |- ( z e. { (/) , { (/) } } <-> ( z = (/) \/ z = { (/) } ) )
19	16, 18	sylib 120	. . . . . . 7 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> ( z = (/) \/ z = { (/) } ) )
20	19	adantl 271	. . . . . 6 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> ( z = (/) \/ z = { (/) } ) )
21	7, 15, 20	mpjaod 670	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
22	21	gen2 1379	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
23		dftr2 3877	. . . 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 144	. . 3 |- Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
25		ssrab2 3079	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } }
26		2ordpr 4267	. . 3 |- Ord { (/) , { (/) } }
27		trssord 4135	. . 3 |- ( ( Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } /\ { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } } /\ Ord { (/) , { (/) } } ) -> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
28	24, 25, 26, 27	mp3an 1268	. 2 |- Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
29		pp0ex 3960	. . . 4 |- { (/) , { (/) } } e. _V
30	29	rabex 3922	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. _V
31	30	elon 4129	. 2 |- ( { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On <-> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
32	28, 31	mpbir 144	1 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   {crab 2352   C_ wss 2973   (/)c0 3251   {csn 3398   {cpr 3399   Tr wtr 3875   Ord word 4117   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by:  onsucelsucexmid  4273  acexmidlemcase  5527  acexmidlemv  5530