MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz9.13 Structured version   Visualization version   GIF version

Theorem tz9.13 9831
Description: Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.)
Hypothesis
Ref Expression
tz9.13.1 𝐴 ∈ V
Assertion
Ref Expression
tz9.13 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem tz9.13
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.13.1 . . 3 𝐴 ∈ V
2 setind 9774 . . . 4 (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V)
3 ssel 3977 . . . . . . . 8 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}))
4 vex 3484 . . . . . . . . 9 𝑤 ∈ V
5 eleq1 2829 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑤 ∈ (𝑅1𝑥)))
65rexbidv 3179 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
74, 6elab 3679 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
83, 7imbitrdi 251 . . . . . . 7 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
98ralrimiv 3145 . . . . . 6 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → ∀𝑤𝑧𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
10 vex 3484 . . . . . . 7 𝑧 ∈ V
1110tz9.12 9830 . . . . . 6 (∀𝑤𝑧𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥) → ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥))
129, 11syl 17 . . . . 5 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥))
13 eleq1 2829 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑧 ∈ (𝑅1𝑥)))
1413rexbidv 3179 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥)))
1510, 14elab 3679 . . . . 5 (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥))
1612, 15sylibr 234 . . . 4 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)})
172, 16mpg 1797 . . 3 {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V
181, 17eleqtrri 2840 . 2 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}
19 eleq1 2829 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1𝑥)))
2019rexbidv 3179 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
211, 20elab 3679 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
2218, 21mpbi 230 1 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951  Oncon0 6384  cfv 6561  𝑅1cr1 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804
This theorem is referenced by:  tz9.13g  9832  elhf2  36176
  Copyright terms: Public domain W3C validator