Theorem setindtrs 42982

Description: Set induction scheme without Infinity. See comments at setindtr 42981. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Hypotheses

Ref	Expression
setindtrs.a	⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
setindtrs.b	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
setindtrs.c	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
setindtrs	⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 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.

Step	Hyp	Ref	Expression
1		setindtr 42981	. . 3 ⊢ (∀𝑎(𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑}) → (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		dfss3 3954	. . . 4 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑})
3		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
4		nfsab1 2720	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5	3, 4	nfralw 3295	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}
6		nfsab1 2720	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 ∈ {𝑥 ∣ 𝜑}
7	5, 6	nfim 1895	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
8		raleq 3307	. . . . . 6 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}))
9		eleq1w 2816	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑎 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑎 → ((∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})))
11		setindtrs.a	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
12		vex 3468	. . . . . . . 8 ⊢ 𝑦 ∈ V
13		setindtrs.b	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
14	12, 13	elab 3663	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
15	14	ralbii 3081	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝜓)
16		abid 2716	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
17	11, 15, 16	3imtr4i 292	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜑})
18	7, 10, 17	chvarfv 2239	. . . 4 ⊢ (∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
19	2, 18	sylbi 217	. . 3 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
20	1, 19	mpg 1796	. 2 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑})
21		elex 3485	. . . . 5 ⊢ (𝐵 ∈ 𝑧 → 𝐵 ∈ V)
22	21	adantl 481	. . . 4 ⊢ ((Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
23	22	exlimiv 1929	. . 3 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
24		setindtrs.c	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
25	24	elabg 3660	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
26	23, 25	syl 17	. 2 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → (𝐵 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
27	20, 26	mpbid 232	1 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  ∀wral 3050  Vcvv 3464   ⊆ wss 3933  Tr wtr 5241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-reg 9615

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-in 3940  df-ss 3950  df-nul 4316  df-uni 4890  df-tr 5242

This theorem is referenced by:  dford3lem2  42984