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Theorem setindtrs 39964
 Description: Set induction scheme without Infinity. See comments at setindtr 39963. (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 39963 . . 3 (∀𝑎(𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑}) → (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑}))
2 dfss3 3903 . . . 4 (𝑎 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑})
3 nfcv 2955 . . . . . . 7 𝑥𝑎
4 nfsab1 2785 . . . . . . 7 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfralw 3189 . . . . . 6 𝑥𝑦𝑎 𝑦 ∈ {𝑥𝜑}
6 nfsab1 2785 . . . . . 6 𝑥 𝑎 ∈ {𝑥𝜑}
75, 6nfim 1897 . . . . 5 𝑥(∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
8 raleq 3358 . . . . . 6 (𝑥 = 𝑎 → (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑}))
9 eleq1w 2872 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑎 ∈ {𝑥𝜑}))
108, 9imbi12d 348 . . . . 5 (𝑥 = 𝑎 → ((∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑}) ↔ (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})))
11 setindtrs.a . . . . . 6 (∀𝑦𝑥 𝜓𝜑)
12 vex 3444 . . . . . . . 8 𝑦 ∈ V
13 setindtrs.b . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1412, 13elab 3615 . . . . . . 7 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
1514ralbii 3133 . . . . . 6 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑥 𝜓)
16 abid 2780 . . . . . 6 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
1711, 15, 163imtr4i 295 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑})
187, 10, 17chvarfv 2240 . . . 4 (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
192, 18sylbi 220 . . 3 (𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
201, 19mpg 1799 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑})
21 elex 3459 . . . . 5 (𝐵𝑧𝐵 ∈ V)
2221adantl 485 . . . 4 ((Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
2322exlimiv 1931 . . 3 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
24 setindtrs.c . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
2524elabg 3614 . . 3 (𝐵 ∈ V → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2623, 25syl 17 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2720, 26mpbid 235 1 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  Tr wtr 5136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-reg 9040 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-uni 4801  df-tr 5137 This theorem is referenced by:  dford3lem2  39966
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