Theorem noextenddif 31821

Description: Calculate the place where a surreal and its extension differ. (Contributed by Scott Fenton, 22-Nov-2021.)

Hypothesis

Ref	Expression
noextend.1	|- X e. { 1o , 2o }

Assertion

Ref	Expression
noextenddif	|- ( A e. No -> |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } = dom A )

Distinct variable groups:   x, A   x, X

Proof of Theorem noextenddif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nodmon 31803	. . 3 |- ( A e. No -> dom A e. On )
2		noextend.1	. . . . . 6 |- X e. { 1o , 2o }
3	2	nosgnn0i 31812	. . . . 5 |- (/) =/= X
4	3	a1i 11	. . . 4 |- ( A e. No -> (/) =/= X )
5		nodmord 31806	. . . . . 6 |- ( A e. No -> Ord dom A )
6		ordirr 5741	. . . . . 6 |- ( Ord dom A -> -. dom A e. dom A )
7	5, 6	syl 17	. . . . 5 |- ( A e. No -> -. dom A e. dom A )
8		ndmfv 6218	. . . . 5 |- ( -. dom A e. dom A -> ( A ` dom A ) = (/) )
9	7, 8	syl 17	. . . 4 |- ( A e. No -> ( A ` dom A ) = (/) )
10		nofun 31802	. . . . . . 7 |- ( A e. No -> Fun A )
11		funfn 5918	. . . . . . 7 |- ( Fun A <-> A Fn dom A )
12	10, 11	sylib 208	. . . . . 6 |- ( A e. No -> A Fn dom A )
13		fnsng 5938	. . . . . . 7 |- ( ( dom A e. On /\ X e. { 1o , 2o } ) -> { <. dom A , X >. } Fn { dom A } )
14	1, 2, 13	sylancl 694	. . . . . 6 |- ( A e. No -> { <. dom A , X >. } Fn { dom A } )
15		disjsn 4246	. . . . . . 7 |- ( ( dom A i^i { dom A } ) = (/) <-> -. dom A e. dom A )
16	7, 15	sylibr 224	. . . . . 6 |- ( A e. No -> ( dom A i^i { dom A } ) = (/) )
17		snidg 4206	. . . . . . 7 |- ( dom A e. On -> dom A e. { dom A } )
18	1, 17	syl 17	. . . . . 6 |- ( A e. No -> dom A e. { dom A } )
19		fvun2 6270	. . . . . 6 |- ( ( A Fn dom A /\ { <. dom A , X >. } Fn { dom A } /\ ( ( dom A i^i { dom A } ) = (/) /\ dom A e. { dom A } ) ) -> ( ( A u. { <. dom A , X >. } ) ` dom A ) = ( { <. dom A , X >. } ` dom A ) )
20	12, 14, 16, 18, 19	syl112anc 1330	. . . . 5 |- ( A e. No -> ( ( A u. { <. dom A , X >. } ) ` dom A ) = ( { <. dom A , X >. } ` dom A ) )
21		fvsng 6447	. . . . . 6 |- ( ( dom A e. On /\ X e. { 1o , 2o } ) -> ( { <. dom A , X >. } ` dom A ) = X )
22	1, 2, 21	sylancl 694	. . . . 5 |- ( A e. No -> ( { <. dom A , X >. } ` dom A ) = X )
23	20, 22	eqtrd 2656	. . . 4 |- ( A e. No -> ( ( A u. { <. dom A , X >. } ) ` dom A ) = X )
24	4, 9, 23	3netr4d 2871	. . 3 |- ( A e. No -> ( A ` dom A ) =/= ( ( A u. { <. dom A , X >. } ) ` dom A ) )
25		fveq2 6191	. . . . 5 |- ( x = dom A -> ( A ` x ) = ( A ` dom A ) )
26		fveq2 6191	. . . . 5 |- ( x = dom A -> ( ( A u. { <. dom A , X >. } ) ` x ) = ( ( A u. { <. dom A , X >. } ) ` dom A ) )
27	25, 26	neeq12d 2855	. . . 4 |- ( x = dom A -> ( ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) <-> ( A ` dom A ) =/= ( ( A u. { <. dom A , X >. } ) ` dom A ) ) )
28	27	onintss 5775	. . 3 |- ( dom A e. On -> ( ( A ` dom A ) =/= ( ( A u. { <. dom A , X >. } ) ` dom A ) -> |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } C_ dom A ) )
29	1, 24, 28	sylc 65	. 2 |- ( A e. No -> |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } C_ dom A )
30		eloni 5733	. . . . . . . 8 |- ( y e. On -> Ord y )
31		ordtri2 5758	. . . . . . . . . 10 |- ( ( Ord y /\ Ord dom A ) -> ( y e. dom A <-> -. ( y = dom A \/ dom A e. y ) ) )
32		eqcom 2629	. . . . . . . . . . . . 13 |- ( y = dom A <-> dom A = y )
33	32	orbi1i 542	. . . . . . . . . . . 12 |- ( ( y = dom A \/ dom A e. y ) <-> ( dom A = y \/ dom A e. y ) )
34		orcom 402	. . . . . . . . . . . 12 |- ( ( dom A = y \/ dom A e. y ) <-> ( dom A e. y \/ dom A = y ) )
35	33, 34	bitri 264	. . . . . . . . . . 11 |- ( ( y = dom A \/ dom A e. y ) <-> ( dom A e. y \/ dom A = y ) )
36	35	notbii 310	. . . . . . . . . 10 |- ( -. ( y = dom A \/ dom A e. y ) <-> -. ( dom A e. y \/ dom A = y ) )
37	31, 36	syl6bb 276	. . . . . . . . 9 |- ( ( Ord y /\ Ord dom A ) -> ( y e. dom A <-> -. ( dom A e. y \/ dom A = y ) ) )
38		ordsseleq 5752	. . . . . . . . . . 11 |- ( ( Ord dom A /\ Ord y ) -> ( dom A C_ y <-> ( dom A e. y \/ dom A = y ) ) )
39	38	notbid 308	. . . . . . . . . 10 |- ( ( Ord dom A /\ Ord y ) -> ( -. dom A C_ y <-> -. ( dom A e. y \/ dom A = y ) ) )
40	39	ancoms 469	. . . . . . . . 9 |- ( ( Ord y /\ Ord dom A ) -> ( -. dom A C_ y <-> -. ( dom A e. y \/ dom A = y ) ) )
41	37, 40	bitr4d 271	. . . . . . . 8 |- ( ( Ord y /\ Ord dom A ) -> ( y e. dom A <-> -. dom A C_ y ) )
42	30, 5, 41	syl2anr 495	. . . . . . 7 |- ( ( A e. No /\ y e. On ) -> ( y e. dom A <-> -. dom A C_ y ) )
43	12	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( A e. No /\ y e. On /\ y e. dom A ) -> A Fn dom A )
44	14	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( A e. No /\ y e. On /\ y e. dom A ) -> { <. dom A , X >. } Fn { dom A } )
45	16	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( A e. No /\ y e. On /\ y e. dom A ) -> ( dom A i^i { dom A } ) = (/) )
46		simp3 1063	. . . . . . . . . 10 |- ( ( A e. No /\ y e. On /\ y e. dom A ) -> y e. dom A )
47		fvun1 6269	. . . . . . . . . 10 |- ( ( A Fn dom A /\ { <. dom A , X >. } Fn { dom A } /\ ( ( dom A i^i { dom A } ) = (/) /\ y e. dom A ) ) -> ( ( A u. { <. dom A , X >. } ) ` y ) = ( A ` y ) )
48	43, 44, 45, 46, 47	syl112anc 1330	. . . . . . . . 9 |- ( ( A e. No /\ y e. On /\ y e. dom A ) -> ( ( A u. { <. dom A , X >. } ) ` y ) = ( A ` y ) )
49	48	eqcomd 2628	. . . . . . . 8 |- ( ( A e. No /\ y e. On /\ y e. dom A ) -> ( A ` y ) = ( ( A u. { <. dom A , X >. } ) ` y ) )
50	49	3expia 1267	. . . . . . 7 |- ( ( A e. No /\ y e. On ) -> ( y e. dom A -> ( A ` y ) = ( ( A u. { <. dom A , X >. } ) ` y ) ) )
51	42, 50	sylbird 250	. . . . . 6 |- ( ( A e. No /\ y e. On ) -> ( -. dom A C_ y -> ( A ` y ) = ( ( A u. { <. dom A , X >. } ) ` y ) ) )
52	51	necon1ad 2811	. . . . 5 |- ( ( A e. No /\ y e. On ) -> ( ( A ` y ) =/= ( ( A u. { <. dom A , X >. } ) ` y ) -> dom A C_ y ) )
53	52	ralrimiva 2966	. . . 4 |- ( A e. No -> A. y e. On ( ( A ` y ) =/= ( ( A u. { <. dom A , X >. } ) ` y ) -> dom A C_ y ) )
54		fveq2 6191	. . . . . 6 |- ( x = y -> ( A ` x ) = ( A ` y ) )
55		fveq2 6191	. . . . . 6 |- ( x = y -> ( ( A u. { <. dom A , X >. } ) ` x ) = ( ( A u. { <. dom A , X >. } ) ` y ) )
56	54, 55	neeq12d 2855	. . . . 5 |- ( x = y -> ( ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) <-> ( A ` y ) =/= ( ( A u. { <. dom A , X >. } ) ` y ) ) )
57	56	ralrab 3368	. . . 4 |- ( A. y e. { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } dom A C_ y <-> A. y e. On ( ( A ` y ) =/= ( ( A u. { <. dom A , X >. } ) ` y ) -> dom A C_ y ) )
58	53, 57	sylibr 224	. . 3 |- ( A e. No -> A. y e. { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } dom A C_ y )
59		ssint 4493	. . 3 |- ( dom A C_ |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } <-> A. y e. { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } dom A C_ y )
60	58, 59	sylibr 224	. 2 |- ( A e. No -> dom A C_ |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } )
61	29, 60	eqssd 3620	1 |- ( A e. No -> |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } = dom A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   |^|cint 4475   dom cdm 5114   Ord word 5722   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796

This theorem is referenced by:  noextendlt  31822  noextendgt  31823