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Theorem tz9.13 9212
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 9168 . . . 4 (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V)
3 ssel 3964 . . . . . . . 8 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}))
4 vex 3502 . . . . . . . . 9 𝑤 ∈ V
5 eleq1 2904 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑤 ∈ (𝑅1𝑥)))
65rexbidv 3301 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
74, 6elab 3670 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
83, 7syl6ib 252 . . . . . . 7 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
98ralrimiv 3185 . . . . . 6 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → ∀𝑤𝑧𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
10 vex 3502 . . . . . . 7 𝑧 ∈ V
1110tz9.12 9211 . . . . . 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 2904 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑧 ∈ (𝑅1𝑥)))
1413rexbidv 3301 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥)))
1510, 14elab 3670 . . . . 5 (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥))
1612, 15sylibr 235 . . . 4 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)})
172, 16mpg 1791 . . 3 {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V
181, 17eleqtrri 2916 . 2 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}
19 eleq1 2904 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1𝑥)))
2019rexbidv 3301 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
211, 20elab 3670 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
2218, 21mpbi 231 1 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2106  {cab 2802  wral 3142  wrex 3143  Vcvv 3499  wss 3939  Oncon0 6188  cfv 6351  𝑅1cr1 9183
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-reg 9048  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-r1 9185
This theorem is referenced by:  tz9.13g  9213  elhf2  33521
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