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Theorem tz9.13g 8511
Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 8510 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
tz9.13g (𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem tz9.13g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2671 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1𝑥)))
21rexbidv 3029 . 2 (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
3 vex 3171 . . 3 𝑦 ∈ V
43tz9.13 8510 . 2 𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥)
52, 4vtoclg 3234 1 (𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  wrex 2892  Oncon0 5622  cfv 5786  𝑅1cr1 8481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-reg 8353  ax-inf2 8394
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-r1 8483
This theorem is referenced by: (None)
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