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Theorem setindtrs 41392
Description: Set induction scheme without Infinity. See comments at setindtr 41391. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Hypotheses
Ref Expression
setindtrs.a (∀𝑦𝑥 𝜓𝜑)
setindtrs.b (𝑥 = 𝑦 → (𝜑𝜓))
setindtrs.c (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
setindtrs (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
Distinct variable groups:   𝑥,𝐵,𝑧   𝜑,𝑦   𝜓,𝑥   𝜒,𝑥   𝜑,𝑧   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝜒(𝑦,𝑧)   𝐵(𝑦)

Proof of Theorem setindtrs
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 setindtr 41391 . . 3 (∀𝑎(𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑}) → (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑}))
2 dfss3 3933 . . . 4 (𝑎 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑})
3 nfcv 2904 . . . . . . 7 𝑥𝑎
4 nfsab1 2718 . . . . . . 7 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfralw 3293 . . . . . 6 𝑥𝑦𝑎 𝑦 ∈ {𝑥𝜑}
6 nfsab1 2718 . . . . . 6 𝑥 𝑎 ∈ {𝑥𝜑}
75, 6nfim 1900 . . . . 5 𝑥(∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
8 raleq 3308 . . . . . 6 (𝑥 = 𝑎 → (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑}))
9 eleq1w 2817 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑎 ∈ {𝑥𝜑}))
108, 9imbi12d 345 . . . . 5 (𝑥 = 𝑎 → ((∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑}) ↔ (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})))
11 setindtrs.a . . . . . 6 (∀𝑦𝑥 𝜓𝜑)
12 vex 3448 . . . . . . . 8 𝑦 ∈ V
13 setindtrs.b . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1412, 13elab 3631 . . . . . . 7 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
1514ralbii 3093 . . . . . 6 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑥 𝜓)
16 abid 2714 . . . . . 6 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
1711, 15, 163imtr4i 292 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑})
187, 10, 17chvarfv 2234 . . . 4 (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
192, 18sylbi 216 . . 3 (𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
201, 19mpg 1800 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑})
21 elex 3462 . . . . 5 (𝐵𝑧𝐵 ∈ V)
2221adantl 483 . . . 4 ((Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
2322exlimiv 1934 . . 3 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
24 setindtrs.c . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
2524elabg 3629 . . 3 (𝐵 ∈ V → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2623, 25syl 17 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2720, 26mpbid 231 1 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  Vcvv 3444  wss 3911  Tr wtr 5223
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-reg 9533
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284  df-uni 4867  df-tr 5224
This theorem is referenced by:  dford3lem2  41394
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