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Theorem wunex3 9136
Description: Construct a weak universe from a given set. This version of wunex 9134 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 8294 . . 3  |-  ( A  e.  V  ->  A  C_  ( R1 `  ( rank `  A ) ) )
2 rankon 8230 . . . . . 6  |-  ( rank `  A )  e.  On
3 omelon 8080 . . . . . 6  |-  om  e.  On
4 oacl 7203 . . . . . 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 6718 . . . . . 6  |-  (/)  e.  om
7 oaord1 7218 . . . . . . 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 8216 . . . . 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 3532 . . 3  |-  ( R1
`  ( rank `  A
) )  C_  U
141, 13syl6ss 3511 . 2  |-  ( A  e.  V  ->  A  C_  U )
15 limom 6714 . . . . . 6  |-  Lim  om
163, 15pm3.2i 455 . . . . 5  |-  ( om  e.  On  /\  Lim  om )
17 oalimcl 7227 . . . . 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 9131 . . . 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 2541 . 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 1395    e. wcel 1819    C_ wss 3471   (/)c0 3793   Oncon0 4887   Lim wlim 4888   ` cfv 5594  (class class class)co 6296   omcom 6699    +o coa 7145   R1cr1 8197   rankcrnk 8198  WUnicwun 9095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-oadd 7152  df-r1 8199  df-rank 8200  df-wun 9097
This theorem is referenced by: (None)
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