Theorem setindtrs 43985

Description: Set induction scheme without Infinity. See comments at setindtr 43984. (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 43984	. . 3 ⊢ (∀𝑎(𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑}) → (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		dfss3 3920	. . . 4 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑})
3		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
4		nfsab1 2747	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5	3, 4	nfralw 3310	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}
6		nfsab1 2747	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 ∈ {𝑥 ∣ 𝜑}
7	5, 6	nfim 1929	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
8		raleq 3317	. . . . . 6 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}))
9		eleq1w 2844	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑎 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	imbi12d 347	. . . . 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 3455	. . . . . . . 8 ⊢ 𝑦 ∈ V
13		setindtrs.b	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
14	12, 13	elab 3633	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
15	14	ralbii 3109	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝜓)
16		abid 2743	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
17	11, 15, 16	3imtr4i 295	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜑})
18	7, 10, 17	chvarfv 2277	. . . 4 ⊢ (∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
19	2, 18	sylbi 220	. . 3 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
20	1, 19	mpg 1830	. 2 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑})
21		elex 3472	. . . . 5 ⊢ (𝐵 ∈ 𝑧 → 𝐵 ∈ V)
22	21	adantl 487	. . . 4 ⊢ ((Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
23	22	exlimiv 1963	. . 3 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
24		setindtrs.c	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
25	24	elabg 3630	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
26	23, 25	syl 18	. 2 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → (𝐵 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
27	20, 26	mpbid 235	1 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  Tr wtr 5212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-reg 9570

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280  df-uni 4868  df-tr 5213

This theorem is used by:  dford3lem2  43987