MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1wunlim Structured version   Visualization version   GIF version

Theorem r1wunlim 10673
Description: The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
r1wunlim (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))

Proof of Theorem r1wunlim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ WUni)
21wun0 10654 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ∅ ∈ (𝑅1𝐴))
3 elfvdm 6879 . . . . . 6 (∅ ∈ (𝑅1𝐴) → 𝐴 ∈ dom 𝑅1)
42, 3syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ dom 𝑅1)
5 r1fnon 9703 . . . . . 6 𝑅1 Fn On
65fndmi 6606 . . . . 5 dom 𝑅1 = On
74, 6eleqtrdi 2848 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ On)
8 eloni 6327 . . . 4 (𝐴 ∈ On → Ord 𝐴)
97, 8syl 17 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Ord 𝐴)
10 n0i 4293 . . . . . 6 (∅ ∈ (𝑅1𝐴) → ¬ (𝑅1𝐴) = ∅)
112, 10syl 17 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) = ∅)
12 fveq2 6842 . . . . . 6 (𝐴 = ∅ → (𝑅1𝐴) = (𝑅1‘∅))
13 r10 9704 . . . . . 6 (𝑅1‘∅) = ∅
1412, 13eqtrdi 2792 . . . . 5 (𝐴 = ∅ → (𝑅1𝐴) = ∅)
1511, 14nsyl 140 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ 𝐴 = ∅)
16 onsuc 7746 . . . . . . . 8 (𝐴 ∈ On → suc 𝐴 ∈ On)
177, 16syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → suc 𝐴 ∈ On)
18 sucidg 6398 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
197, 18syl 17 . . . . . . 7 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → 𝐴 ∈ suc 𝐴)
20 r1ord 9716 . . . . . . 7 (suc 𝐴 ∈ On → (𝐴 ∈ suc 𝐴 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴)))
2117, 19, 20sylc 65 . . . . . 6 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
22 r1elwf 9732 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
23 wfelirr 9761 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1 “ On) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
2421, 22, 233syl 18 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝑅1𝐴) ∈ (𝑅1𝐴))
25 simprr 771 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 = suc 𝑥)
2625fveq2d 6846 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = (𝑅1‘suc 𝑥))
27 r1suc 9706 . . . . . . . . 9 (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2827ad2antrl 726 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2926, 28eqtrd 2776 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) = 𝒫 (𝑅1𝑥))
30 simplr 767 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ WUni)
317adantr 481 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝐴 ∈ On)
32 sucidg 6398 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥 ∈ suc 𝑥)
3332ad2antrl 726 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥 ∈ suc 𝑥)
3433, 25eleqtrrd 2841 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝑥𝐴)
35 r1ord 9716 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴 → (𝑅1𝑥) ∈ (𝑅1𝐴)))
3631, 34, 35sylc 65 . . . . . . . 8 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝑥) ∈ (𝑅1𝐴))
3730, 36wunpw 10643 . . . . . . 7 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → 𝒫 (𝑅1𝑥) ∈ (𝑅1𝐴))
3829, 37eqeltrd 2838 . . . . . 6 (((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) ∧ (𝑥 ∈ On ∧ 𝐴 = suc 𝑥)) → (𝑅1𝐴) ∈ (𝑅1𝐴))
3938rexlimdvaa 3153 . . . . 5 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝑅1𝐴) ∈ (𝑅1𝐴)))
4024, 39mtod 197 . . . 4 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
41 ioran 982 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
4215, 40, 41sylanbrc 583 . . 3 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
43 dflim3 7783 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
449, 42, 43sylanbrc 583 . 2 ((𝐴𝑉 ∧ (𝑅1𝐴) ∈ WUni) → Lim 𝐴)
45 r1limwun 10672 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
4644, 45impbida 799 1 (𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wrex 3073  c0 4282  𝒫 cpw 4560   cuni 4865  dom cdm 5633  cima 5636  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  cfv 6496  𝑅1cr1 9698  WUnicwun 10636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-r1 9700  df-rank 9701  df-wun 10638
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator