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Theorem tz9.13 8825
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 8781 . . . 4 (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V)
3 ssel 3736 . . . . . . . 8 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}))
4 vex 3341 . . . . . . . . 9 𝑤 ∈ V
5 eleq1 2825 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑤 ∈ (𝑅1𝑥)))
65rexbidv 3188 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
74, 6elab 3488 . . . . . . . 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 3101 . . . . . 6 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → ∀𝑤𝑧𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
10 vex 3341 . . . . . . 7 𝑧 ∈ V
1110tz9.12 8824 . . . . . 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 2825 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑧 ∈ (𝑅1𝑥)))
1413rexbidv 3188 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥)))
1510, 14elab 3488 . . . . 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 1871 . . 3 {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V
181, 17eleqtrri 2836 . 2 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}
19 eleq1 2825 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1𝑥)))
2019rexbidv 3188 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
211, 20elab 3488 . 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 1630  wcel 2137  {cab 2744  wral 3048  wrex 3049  Vcvv 3338  wss 3713  Oncon0 5882  cfv 6047  𝑅1cr1 8796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-reg 8660  ax-inf2 8709
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-om 7229  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-r1 8798
This theorem is referenced by:  tz9.13g  8826  elhf2  32586
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