Theorem setindtrs 43614

Description: Set induction scheme without Infinity. See comments at setindtr 43613. (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 43613	. . 3 ⊢ (∀𝑎(𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑}) → (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		dfss3 3928	. . . 4 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑})
3		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
4		nfsab1 2751	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5	3, 4	nfralw 3312	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}
6		nfsab1 2751	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 ∈ {𝑥 ∣ 𝜑}
7	5, 6	nfim 1919	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
8		raleq 3320	. . . . . 6 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}))
9		eleq1w 2848	. . . . . 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 3461	. . . . . . . 8 ⊢ 𝑦 ∈ V
13		setindtrs.b	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
14	12, 13	elab 3641	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
15	14	ralbii 3111	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝜓)
16		abid 2747	. . . . . 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 2278	. . . 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 1820	. 2 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑})
21		elex 3478	. . . . 5 ⊢ (𝐵 ∈ 𝑧 → 𝐵 ∈ V)
22	21	adantl 486	. . . 4 ⊢ ((Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
23	22	exlimiv 1953	. . 3 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
24		setindtrs.c	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
25	24	elabg 3638	. . 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  Tr wtr 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-reg 9542

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-in 3914  df-ss 3924  df-nul 4289  df-uni 4869  df-tr 5213

This theorem is referenced by:  dford3lem2  43616