Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunex3 Structured version   Visualization version   GIF version

Theorem wunex3 10155
 Description: Construct a weak universe from a given set. This version of wunex 10153 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 ω))
Assertion
Ref Expression
wunex3 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))

Proof of Theorem wunex3
StepHypRef Expression
1 r1rankid 9280 . . 3 (𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 rankon 9216 . . . . . 6 (rank‘𝐴) ∈ On
3 omelon 9101 . . . . . 6 ω ∈ On
4 oacl 8152 . . . . . 6 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On)
52, 3, 4mp2an 690 . . . . 5 ((rank‘𝐴) +o ω) ∈ On
6 peano1 7593 . . . . . 6 ∅ ∈ ω
7 oaord1 8169 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)))
82, 3, 7mp2an 690 . . . . . 6 (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))
96, 8mpbi 232 . . . . 5 (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)
10 r1ord2 9202 . . . . 5 (((rank‘𝐴) +o ω) ∈ On → ((rank‘𝐴) ∈ ((rank‘𝐴) +o ω) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) +o ω))))
115, 9, 10mp2 9 . . . 4 (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) +o ω))
12 wunex3.u . . . 4 𝑈 = (𝑅1‘((rank‘𝐴) +o ω))
1311, 12sseqtrri 4002 . . 3 (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
141, 13sstrdi 3977 . 2 (𝐴𝑉𝐴𝑈)
15 limom 7587 . . . . . 6 Lim ω
163, 15pm3.2i 473 . . . . 5 (ω ∈ On ∧ Lim ω)
17 oalimcl 8178 . . . . 5 (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω))
182, 16, 17mp2an 690 . . . 4 Lim ((rank‘𝐴) +o ω)
19 r1limwun 10150 . . . 4 ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni)
205, 18, 19mp2an 690 . . 3 (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni
2112, 20eqeltri 2907 . 2 𝑈 ∈ WUni
2214, 21jctil 522 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1530   ∈ wcel 2107   ⊆ wss 3934  ∅c0 4289  Oncon0 6184  Lim wlim 6185  ‘cfv 6348  (class class class)co 7148  ωcom 7572   +o coa 8091  𝑅1cr1 9183  rankcrnk 9184  WUnicwun 10114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-reg 9048  ax-inf2 9096 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-oadd 8098  df-r1 9185  df-rank 9186  df-wun 10116 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator