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Theorem noextendseq 31820
Description: Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.)
Hypothesis
Ref Expression
noextend.1 |- X e. { 1o , 2o }
Assertion
Ref Expression
noextendseq |- ( ( A e. No /\ B e. On ) -> ( A u. ( ( B \ dom A ) X. { X } ) ) e. No )
Proof of Theorem noextendseq
StepHypRef Expression
1 nofun 31802 . . . 4 |- ( A e. No -> Fun A )
2 noextend.1 . . . . . 6 |- X e. { 1o , 2o }
3 fnconstg 6093 . . . . . 6 |- ( X e. { 1o , 2o } -> ( ( B \ dom A ) X. { X } ) Fn ( B \ dom A ) )
4 fnfun 5988 . . . . . 6 |- ( ( ( B \ dom A ) X. { X } ) Fn ( B \ dom A ) -> Fun ( ( B \ dom A ) X. { X } ) )
52, 3, 4mp2b 10 . . . . 5 |- Fun ( ( B \ dom A ) X. { X } )
6 snnzg 4308 . . . . . . . . 9 |- ( X e. { 1o , 2o } -> { X } =/= (/) )
7 dmxp 5344 . . . . . . . . 9 |- ( { X } =/= (/) -> dom ( ( B \ dom A ) X. { X } ) = ( B \ dom A ) )
82, 6, 7mp2b 10 . . . . . . . 8 |- dom ( ( B \ dom A ) X. { X } ) = ( B \ dom A )
98ineq2i 3811 . . . . . . 7 |- ( dom A i^i dom ( ( B \ dom A ) X. { X } ) ) = ( dom A i^i ( B \ dom A ) )
10 disjdif 4040 . . . . . . 7 |- ( dom A i^i ( B \ dom A ) ) = (/)
119, 10eqtri 2644 . . . . . 6 |- ( dom A i^i dom ( ( B \ dom A ) X. { X } ) ) = (/)
12 funun 5932 . . . . . 6 |- ( ( ( Fun A /\ Fun ( ( B \ dom A ) X. { X } ) ) /\ ( dom A i^i dom ( ( B \ dom A ) X. { X } ) ) = (/) ) -> Fun ( A u. ( ( B \ dom A ) X. { X } ) ) )
1311, 12mpan2 707 . . . . 5 |- ( ( Fun A /\ Fun ( ( B \ dom A ) X. { X } ) ) -> Fun ( A u. ( ( B \ dom A ) X. { X } ) ) )
145, 13mpan2 707 . . . 4 |- ( Fun A -> Fun ( A u. ( ( B \ dom A ) X. { X } ) ) )
151, 14syl 17 . . 3 |- ( A e. No -> Fun ( A u. ( ( B \ dom A ) X. { X } ) ) )
1615adantr 481 . 2 |- ( ( A e. No /\ B e. On ) -> Fun ( A u. ( ( B \ dom A ) X. { X } ) ) )
17 dmun 5331 . . . 4 |- dom ( A u. ( ( B \ dom A ) X. { X } ) ) = ( dom A u. dom ( ( B \ dom A ) X. { X } ) )
188uneq2i 3764 . . . 4 |- ( dom A u. dom ( ( B \ dom A ) X. { X } ) ) = ( dom A u. ( B \ dom A ) )
1917, 18eqtri 2644 . . 3 |- dom ( A u. ( ( B \ dom A ) X. { X } ) ) = ( dom A u. ( B \ dom A ) )
20 nodmon 31803 . . . 4 |- ( A e. No -> dom A e. On )
21 undif 4049 . . . . . 6 |- ( dom A C_ B <-> ( dom A u. ( B \ dom A ) ) = B )
22 eleq1a 2696 . . . . . . 7 |- ( B e. On -> ( ( dom A u. ( B \ dom A ) ) = B -> ( dom A u. ( B \ dom A ) ) e. On ) )
2322adantl 482 . . . . . 6 |- ( ( dom A e. On /\ B e. On ) -> ( ( dom A u. ( B \ dom A ) ) = B -> ( dom A u. ( B \ dom A ) ) e. On ) )
2421, 23syl5bi 232 . . . . 5 |- ( ( dom A e. On /\ B e. On ) -> ( dom A C_ B -> ( dom A u. ( B \ dom A ) ) e. On ) )
25 ssdif0 3942 . . . . . 6 |- ( B C_ dom A <-> ( B \ dom A ) = (/) )
26 uneq2 3761 . . . . . . . . . 10 |- ( ( B \ dom A ) = (/) -> ( dom A u. ( B \ dom A ) ) = ( dom A u. (/) ) )
27 un0 3967 . . . . . . . . . 10 |- ( dom A u. (/) ) = dom A
2826, 27syl6eq 2672 . . . . . . . . 9 |- ( ( B \ dom A ) = (/) -> ( dom A u. ( B \ dom A ) ) = dom A )
2928eleq1d 2686 . . . . . . . 8 |- ( ( B \ dom A ) = (/) -> ( ( dom A u. ( B \ dom A ) ) e. On <-> dom A e. On ) )
3029biimprcd 240 . . . . . . 7 |- ( dom A e. On -> ( ( B \ dom A ) = (/) -> ( dom A u. ( B \ dom A ) ) e. On ) )
3130adantr 481 . . . . . 6 |- ( ( dom A e. On /\ B e. On ) -> ( ( B \ dom A ) = (/) -> ( dom A u. ( B \ dom A ) ) e. On ) )
3225, 31syl5bi 232 . . . . 5 |- ( ( dom A e. On /\ B e. On ) -> ( B C_ dom A -> ( dom A u. ( B \ dom A ) ) e. On ) )
33 eloni 5733 . . . . . 6 |- ( dom A e. On -> Ord dom A )
34 eloni 5733 . . . . . 6 |- ( B e. On -> Ord B )
35 ordtri2or2 5823 . . . . . 6 |- ( ( Ord dom A /\ Ord B ) -> ( dom A C_ B \/ B C_ dom A ) )
3633, 34, 35syl2an 494 . . . . 5 |- ( ( dom A e. On /\ B e. On ) -> ( dom A C_ B \/ B C_ dom A ) )
3724, 32, 36mpjaod 396 . . . 4 |- ( ( dom A e. On /\ B e. On ) -> ( dom A u. ( B \ dom A ) ) e. On )
3820, 37sylan 488 . . 3 |- ( ( A e. No /\ B e. On ) -> ( dom A u. ( B \ dom A ) ) e. On )
3919, 38syl5eqel 2705 . 2 |- ( ( A e. No /\ B e. On ) -> dom ( A u. ( ( B \ dom A ) X. { X } ) ) e. On )
40 rnun 5541 . . 3 |- ran ( A u. ( ( B \ dom A ) X. { X } ) ) = ( ran A u. ran ( ( B \ dom A ) X. { X } ) )
41 norn 31804 . . . . 5 |- ( A e. No -> ran A C_ { 1o , 2o } )
4241adantr 481 . . . 4 |- ( ( A e. No /\ B e. On ) -> ran A C_ { 1o , 2o } )
43 rnxpss 5566 . . . . 5 |- ran ( ( B \ dom A ) X. { X } ) C_ { X }
44 snssi 4339 . . . . . 6 |- ( X e. { 1o , 2o } -> { X } C_ { 1o , 2o } )
452, 44ax-mp 5 . . . . 5 |- { X } C_ { 1o , 2o }
4643, 45sstri 3612 . . . 4 |- ran ( ( B \ dom A ) X. { X } ) C_ { 1o , 2o }
47 unss 3787 . . . 4 |- ( ( ran A C_ { 1o , 2o } /\ ran ( ( B \ dom A ) X. { X } ) C_ { 1o , 2o } ) <-> ( ran A u. ran ( ( B \ dom A ) X. { X } ) ) C_ { 1o , 2o } )
4842, 46, 47sylanblc 696 . . 3 |- ( ( A e. No /\ B e. On ) -> ( ran A u. ran ( ( B \ dom A ) X. { X } ) ) C_ { 1o , 2o } )
4940, 48syl5eqss 3649 . 2 |- ( ( A e. No /\ B e. On ) -> ran ( A u. ( ( B \ dom A ) X. { X } ) ) C_ { 1o , 2o } )
50 elno2 31807 . 2 |- ( ( A u. ( ( B \ dom A ) X. { X } ) ) e. No <-> ( Fun ( A u. ( ( B \ dom A ) X. { X } ) ) /\ dom ( A u. ( ( B \ dom A ) X. { X } ) ) e. On /\ ran ( A u. ( ( B \ dom A ) X. { X } ) ) C_ { 1o , 2o } ) )
5116, 39, 49, 50syl3anbrc 1246 1 |- ( ( A e. No /\ B e. On ) -> ( A u. ( ( B \ dom A ) X. { X } ) ) e. No )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   X. cxp 5112   dom cdm 5114   ran crn 5115   Ord word 5722   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-no 31796
This theorem is referenced by:  noetalem1  31863
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