Theorem setindtr 41748

Description: Set induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 9725; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
setindtr	⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → 𝐵 ∈ 𝐴))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem setindtr

Step	Hyp	Ref	Expression
1		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Tr 𝑦
2		nfa1 2148	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)
3	1, 2	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
4		eldifn 4126	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
5	4	adantl 482	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
6		trss 5275	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
7		eldifi 4125	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → 𝑥 ∈ 𝑦)
8	6, 7	impel 506	. . . . . . . . . . . . . . . . 17 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → 𝑥 ⊆ 𝑦)
9		df-ss 3964	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
10	8, 9	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
11	10	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
12	11	sseq1d 4012	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		sp 2176	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
14	13	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
15	12, 14	sylbid 239	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 → 𝑥 ∈ 𝐴))
16	5, 15	mtod 197	. . . . . . . . . . . 12 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ 𝑦) ⊆ 𝐴)
17		inssdif0 4368	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
18	16, 17	sylnib 327	. . . . . . . . . . 11 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
19	18	ex 413	. . . . . . . . . 10 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅))
20	3, 19	ralrimi 3254	. . . . . . . . 9 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
21		ralnex 3072	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
22	20, 21	sylib 217	. . . . . . . 8 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
23		vex 3478	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	23	difexi 5327	. . . . . . . . . 10 ⊢ (𝑦 ∖ 𝐴) ∈ V
25		zfreg 9586	. . . . . . . . . 10 ⊢ (((𝑦 ∖ 𝐴) ∈ V ∧ (𝑦 ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
26	24, 25	mpan 688	. . . . . . . . 9 ⊢ ((𝑦 ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
27	26	necon1bi 2969	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅ → (𝑦 ∖ 𝐴) = ∅)
28	22, 27	syl 17	. . . . . . 7 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑦 ∖ 𝐴) = ∅)
29		ssdif0 4362	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝑦 ∖ 𝐴) = ∅)
30	28, 29	sylibr 233	. . . . . 6 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
31	30	adantlr 713	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
32		simplr 767	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑦)
33	31, 32	sseldd 3982	. . . 4 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
34	33	ex 413	. . 3 ⊢ ((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴))
35	34	exlimiv 1933	. 2 ⊢ (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴))
36	35	com12 32	1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  Tr wtr 5264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-reg 9583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322  df-uni 4908  df-tr 5265

This theorem is referenced by:  setindtrs  41749