Theorem nolt02olem 31844

Description: Lemma for nolt02o 31845. If A ( X ) is undefined with A surreal and X ordinal, then dom A C_ X. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nolt02olem	|- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> dom A C_ X )

Proof of Theorem nolt02olem

Step	Hyp	Ref	Expression
1		nosgnn0 31811	. . . 4 |- -. (/) e. { 1o , 2o }
2	1	a1i 11	. . 3 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> -. (/) e. { 1o , 2o } )
3		simpl3 1066	. . . 4 |- ( ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) /\ X e. dom A ) -> ( A ` X ) = (/) )
4		simpl1 1064	. . . . . 6 |- ( ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) /\ X e. dom A ) -> A e. No )
5		norn 31804	. . . . . 6 |- ( A e. No -> ran A C_ { 1o , 2o } )
6	4, 5	syl 17	. . . . 5 |- ( ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) /\ X e. dom A ) -> ran A C_ { 1o , 2o } )
7		nofun 31802	. . . . . . 7 |- ( A e. No -> Fun A )
8	7	3ad2ant1 1082	. . . . . 6 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> Fun A )
9		fvelrn 6352	. . . . . 6 |- ( ( Fun A /\ X e. dom A ) -> ( A ` X ) e. ran A )
10	8, 9	sylan 488	. . . . 5 |- ( ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) /\ X e. dom A ) -> ( A ` X ) e. ran A )
11	6, 10	sseldd 3604	. . . 4 |- ( ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) /\ X e. dom A ) -> ( A ` X ) e. { 1o , 2o } )
12	3, 11	eqeltrrd 2702	. . 3 |- ( ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) /\ X e. dom A ) -> (/) e. { 1o , 2o } )
13	2, 12	mtand 691	. 2 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> -. X e. dom A )
14		nodmon 31803	. . . 4 |- ( A e. No -> dom A e. On )
15	14	3ad2ant1 1082	. . 3 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> dom A e. On )
16		simp2 1062	. . 3 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> X e. On )
17		ontri1 5757	. . 3 |- ( ( dom A e. On /\ X e. On ) -> ( dom A C_ X <-> -. X e. dom A ) )
18	15, 16, 17	syl2anc 693	. 2 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> ( dom A C_ X <-> -. X e. dom A ) )
19	13, 18	mpbird 247	1 |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> dom A C_ X )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   {cpr 4179   dom cdm 5114   ran crn 5115   Oncon0 5723   Fun wfun 5882   ` 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-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-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-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:  nolt02o  31845