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Theorem tz9.13 8605
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 8561 . . . 4 (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V)
3 ssel 3581 . . . . . . . 8 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}))
4 vex 3192 . . . . . . . . 9 𝑤 ∈ V
5 eleq1 2686 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑤 ∈ (𝑅1𝑥)))
65rexbidv 3046 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
74, 6elab 3337 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
83, 7syl6ib 241 . . . . . . 7 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
98ralrimiv 2960 . . . . . 6 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → ∀𝑤𝑧𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
10 vex 3192 . . . . . . 7 𝑧 ∈ V
1110tz9.12 8604 . . . . . 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 2686 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑧 ∈ (𝑅1𝑥)))
1413rexbidv 3046 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥)))
1510, 14elab 3337 . . . . 5 (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥))
1612, 15sylibr 224 . . . 4 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)})
172, 16mpg 1721 . . 3 {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V
181, 17eleqtrri 2697 . 2 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}
19 eleq1 2686 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1𝑥)))
2019rexbidv 3046 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
211, 20elab 3337 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
2218, 21mpbi 220 1 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189  wss 3559  Oncon0 5687  cfv 5852  𝑅1cr1 8576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8448  ax-inf2 8489
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8578
This theorem is referenced by:  tz9.13g  8606  elhf2  31951
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