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Theorem tz9.13 9549
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 9492 . . . 4 (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V)
3 ssel 3914 . . . . . . . 8 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}))
4 vex 3436 . . . . . . . . 9 𝑤 ∈ V
5 eleq1 2826 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑤 ∈ (𝑅1𝑥)))
65rexbidv 3226 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
74, 6elab 3609 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
83, 7syl6ib 250 . . . . . . 7 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → (𝑤𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥)))
98ralrimiv 3102 . . . . . 6 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → ∀𝑤𝑧𝑥 ∈ On 𝑤 ∈ (𝑅1𝑥))
10 vex 3436 . . . . . . 7 𝑧 ∈ V
1110tz9.12 9548 . . . . . 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 2826 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝑧 ∈ (𝑅1𝑥)))
1413rexbidv 3226 . . . . . 6 (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥)))
1510, 14elab 3609 . . . . 5 (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1𝑥))
1612, 15sylibr 233 . . . 4 (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)})
172, 16mpg 1800 . . 3 {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} = V
181, 17eleqtrri 2838 . 2 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)}
19 eleq1 2826 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1𝑥)))
2019rexbidv 3226 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
211, 20elab 3609 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
2218, 21mpbi 229 1 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wrex 3065  Vcvv 3432  wss 3887  Oncon0 6266  cfv 6433  𝑅1cr1 9520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522
This theorem is referenced by:  tz9.13g  9550  elhf2  34477
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