Theorem onsucelsucexmidlem 4530

Description: Lemma for onsucelsucexmid 4531. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5869), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4520. (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 3428	. . . . . . . . . 10 |- -. y e. (/)
3		eleq2 2241	. . . . . . . . . 10 |- ( z = (/) -> ( y e. z <-> y e. (/) ) )
4	2, 3	mtbiri 675	. . . . . . . . 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 620	. . . . . . 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 2241	. . . . . . . . . . 11 |- ( z = { (/) } -> ( y e. z <-> y e. { (/) } ) )
9	8	biimpac 298	. . . . . . . . . 10 |- ( ( y e. z /\ z = { (/) } ) -> y e. { (/) } )
10		velsn 3611	. . . . . . . . . 10 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	sylib 122	. . . . . . . . 9 |- ( ( y e. z /\ z = { (/) } ) -> y = (/) )
12		onsucelsucexmidlem1 4529	. . . . . . . . 9 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
13	11, 12	eqeltrdi 2268	. . . . . . . 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 2892	. . . . . . . 8 |- ( z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } -> z e. { (/) , { (/) } } )
17		vex 2742	. . . . . . . . 9 |- z e. _V
18	17	elpr 3615	. . . . . . . 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 718	. . . . 5 |- ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
22	21	gen2 1450	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } ) -> y e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
23		dftr2 4105	. . . 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 3242	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } }
26		2ordpr 4525	. . 3 |- Ord { (/) , { (/) } }
27		trssord 4382	. . 3 |- ( ( Tr { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } /\ { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } C_ { (/) , { (/) } } /\ Ord { (/) , { (/) } } ) -> Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } )
28	24, 25, 26, 27	mp3an 1337	. 2 |- Ord { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
29		pp0ex 4191	. . . 4 |- { (/) , { (/) } } e. _V
30	29	rabex 4149	. . 3 |- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. _V
31	30	elon 4376	. 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 708   A.wal 1351   = wceq 1353   e. wcel 2148   {crab 2459   C_ wss 3131   (/)c0 3424   {csn 3594   {cpr 3595   Tr wtr 4103   Ord word 4364   Oncon0 4365

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by:  onsucelsucexmid  4531  acexmidlemcase  5873  acexmidlemv  5876