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Theorem wunex3 9115
Description: Construct a weak universe from a given set. This version of wunex 9113 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypothesis
Ref Expression
wunex3.u  |-  U  =  ( R1 `  (
( rank `  A )  +o  om ) )
Assertion
Ref Expression
wunex3  |-  ( A  e.  V  ->  ( U  e. WUni  /\  A  C_  U ) )

Proof of Theorem wunex3
StepHypRef Expression
1 r1rankid 8273 . . 3  |-  ( A  e.  V  ->  A  C_  ( R1 `  ( rank `  A ) ) )
2 rankon 8209 . . . . . 6  |-  ( rank `  A )  e.  On
3 omelon 8059 . . . . . 6  |-  om  e.  On
4 oacl 7182 . . . . . 6  |-  ( ( ( rank `  A
)  e.  On  /\  om  e.  On )  -> 
( ( rank `  A
)  +o  om )  e.  On )
52, 3, 4mp2an 672 . . . . 5  |-  ( (
rank `  A )  +o  om )  e.  On
6 peano1 6697 . . . . . 6  |-  (/)  e.  om
7 oaord1 7197 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  om  e.  On )  -> 
( (/)  e.  om  <->  ( rank `  A )  e.  ( ( rank `  A
)  +o  om )
) )
82, 3, 7mp2an 672 . . . . . 6  |-  ( (/)  e.  om  <->  ( rank `  A
)  e.  ( (
rank `  A )  +o  om ) )
96, 8mpbi 208 . . . . 5  |-  ( rank `  A )  e.  ( ( rank `  A
)  +o  om )
10 r1ord2 8195 . . . . 5  |-  ( ( ( rank `  A
)  +o  om )  e.  On  ->  ( ( rank `  A )  e.  ( ( rank `  A
)  +o  om )  ->  ( R1 `  ( rank `  A ) ) 
C_  ( R1 `  ( ( rank `  A
)  +o  om )
) ) )
115, 9, 10mp2 9 . . . 4  |-  ( R1
`  ( rank `  A
) )  C_  ( R1 `  ( ( rank `  A )  +o  om ) )
12 wunex3.u . . . 4  |-  U  =  ( R1 `  (
( rank `  A )  +o  om ) )
1311, 12sseqtr4i 3537 . . 3  |-  ( R1
`  ( rank `  A
) )  C_  U
141, 13syl6ss 3516 . 2  |-  ( A  e.  V  ->  A  C_  U )
15 limom 6693 . . . . . 6  |-  Lim  om
163, 15pm3.2i 455 . . . . 5  |-  ( om  e.  On  /\  Lim  om )
17 oalimcl 7206 . . . . 5  |-  ( ( ( rank `  A
)  e.  On  /\  ( om  e.  On  /\  Lim  om ) )  ->  Lim  ( ( rank `  A
)  +o  om )
)
182, 16, 17mp2an 672 . . . 4  |-  Lim  (
( rank `  A )  +o  om )
19 r1limwun 9110 . . . 4  |-  ( ( ( ( rank `  A
)  +o  om )  e.  On  /\  Lim  (
( rank `  A )  +o  om ) )  -> 
( R1 `  (
( rank `  A )  +o  om ) )  e. WUni
)
205, 18, 19mp2an 672 . . 3  |-  ( R1
`  ( ( rank `  A )  +o  om ) )  e. WUni
2112, 20eqeltri 2551 . 2  |-  U  e. WUni
2214, 21jctil 537 1  |-  ( A  e.  V  ->  ( U  e. WUni  /\  A  C_  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   (/)c0 3785   Oncon0 4878   Lim wlim 4879   ` cfv 5586  (class class class)co 6282   omcom 6678    +o coa 7124   R1cr1 8176   rankcrnk 8177  WUnicwun 9074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-reg 8014  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-oadd 7131  df-r1 8178  df-rank 8179  df-wun 9076
This theorem is referenced by: (None)
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