Theorem setindtrs 36610

Description: Epsilon induction scheme without Infinity. See comments at setindtr 36609. (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 36609	. . 3 ⊢ (∀𝑎(𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑}) → (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		dfss3 3558	. . . 4 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑})
3		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
4		nfsab1 2600	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
5	3, 4	nfral 2929	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}
6		nfsab1 2600	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 ∈ {𝑥 ∣ 𝜑}
7	5, 6	nfim 1813	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
8		raleq 3115	. . . . . 6 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑}))
9		eleq1 2676	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑎 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	imbi12d 333	. . . . 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 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
13		setindtrs.b	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
14	12, 13	elab 3319	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
15	14	ralbii 2963	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝜓)
16		abid 2598	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
17	11, 15, 16	3imtr4i 280	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜑})
18	7, 10, 17	chvar 2250	. . . 4 ⊢ (∀𝑦 ∈ 𝑎 𝑦 ∈ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
19	2, 18	sylbi 206	. . 3 ⊢ (𝑎 ⊆ {𝑥 ∣ 𝜑} → 𝑎 ∈ {𝑥 ∣ 𝜑})
20	1, 19	mpg 1715	. 2 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ {𝑥 ∣ 𝜑})
21		elex 3185	. . . . 5 ⊢ (𝐵 ∈ 𝑧 → 𝐵 ∈ V)
22	21	adantl 481	. . . 4 ⊢ ((Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
23	22	exlimiv 1845	. . 3 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝐵 ∈ V)
24		setindtrs.c	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
25	24	elabg 3320	. . 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 221	1 ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  Tr wtr 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-uni 4373  df-tr 4681

This theorem is referenced by:  dford3lem2  36612