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Theorem wunex3 8913
Description: Construct a weak universe from a given set. This version of wunex 8911 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 8071 . . 3  |-  ( A  e.  V  ->  A  C_  ( R1 `  ( rank `  A ) ) )
2 rankon 8007 . . . . . 6  |-  ( rank `  A )  e.  On
3 omelon 7857 . . . . . 6  |-  om  e.  On
4 oacl 6980 . . . . . 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 6500 . . . . . 6  |-  (/)  e.  om
7 oaord1 6995 . . . . . . 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 7993 . . . . 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 3394 . . 3  |-  ( R1
`  ( rank `  A
) )  C_  U
141, 13syl6ss 3373 . 2  |-  ( A  e.  V  ->  A  C_  U )
15 limom 6496 . . . . . 6  |-  Lim  om
163, 15pm3.2i 455 . . . . 5  |-  ( om  e.  On  /\  Lim  om )
17 oalimcl 7004 . . . . 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 8908 . . . 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 2513 . 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 1369    e. wcel 1756    C_ wss 3333   (/)c0 3642   Oncon0 4724   Lim wlim 4725   ` cfv 5423  (class class class)co 6096   omcom 6481    +o coa 6922   R1cr1 7974   rankcrnk 7975  WUnicwun 8872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-oadd 6929  df-r1 7976  df-rank 7977  df-wun 8874
This theorem is referenced by: (None)
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