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Theorem setindtrs 39500
Description: Set induction scheme without Infinity. See comments at setindtr 39499. (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 39499 . . 3 (∀𝑎(𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑}) → (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑}))
2 dfss3 3953 . . . 4 (𝑎 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑})
3 nfcv 2974 . . . . . . 7 𝑥𝑎
4 nfsab1 2805 . . . . . . 7 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfralw 3222 . . . . . 6 𝑥𝑦𝑎 𝑦 ∈ {𝑥𝜑}
6 nfsab1 2805 . . . . . 6 𝑥 𝑎 ∈ {𝑥𝜑}
75, 6nfim 1888 . . . . 5 𝑥(∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
8 raleq 3403 . . . . . 6 (𝑥 = 𝑎 → (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑}))
9 eleq1w 2892 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑎 ∈ {𝑥𝜑}))
108, 9imbi12d 346 . . . . 5 (𝑥 = 𝑎 → ((∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑}) ↔ (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})))
11 setindtrs.a . . . . . 6 (∀𝑦𝑥 𝜓𝜑)
12 vex 3495 . . . . . . . 8 𝑦 ∈ V
13 setindtrs.b . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1412, 13elab 3664 . . . . . . 7 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
1514ralbii 3162 . . . . . 6 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑥 𝜓)
16 abid 2800 . . . . . 6 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
1711, 15, 163imtr4i 293 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑})
187, 10, 17chvarfv 2232 . . . 4 (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
192, 18sylbi 218 . . 3 (𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
201, 19mpg 1789 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑})
21 elex 3510 . . . . 5 (𝐵𝑧𝐵 ∈ V)
2221adantl 482 . . . 4 ((Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
2322exlimiv 1922 . . 3 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
24 setindtrs.c . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
2524elabg 3663 . . 3 (𝐵 ∈ V → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2623, 25syl 17 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2720, 26mpbid 233 1 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  Vcvv 3492  wss 3933  Tr wtr 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-uni 4831  df-tr 5164
This theorem is referenced by:  dford3lem2  39502
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