Theorem setindtr 43811

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 9723; 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 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Tr 𝑦
2		nfa1 2189	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)
3	1, 2	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
4		eldifn 4086	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
5	4	adantl 487	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
6		trss 5230	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
7		eldifi 4085	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → 𝑥 ∈ 𝑦)
8	6, 7	impel 515	. . . . . . . . . . . . . . . . 17 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → 𝑥 ⊆ 𝑦)
9		dfss2 3924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
10	8, 9	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
11	10	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
12	11	sseq1d 3969	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		sp 2222	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
14	13	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
15	12, 14	sylbid 243	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 → 𝑥 ∈ 𝐴))
16	5, 15	mtod 201	. . . . . . . . . . . 12 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ 𝑦) ⊆ 𝐴)
17		inssdif0 4329	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
18	16, 17	sylnib 331	. . . . . . . . . . 11 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
19	18	ex 418	. . . . . . . . . 10 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅))
20	3, 19	ralrimi 3265	. . . . . . . . 9 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
21		ralnex 3093	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
22	20, 21	sylib 221	. . . . . . . 8 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
23		vex 3461	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	23	difexi 5303	. . . . . . . . . 10 ⊢ (𝑦 ∖ 𝐴) ∈ V
25		zfreg 9565	. . . . . . . . . 10 ⊢ (((𝑦 ∖ 𝐴) ∈ V ∧ (𝑦 ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
26	24, 25	mpan 703	. . . . . . . . 9 ⊢ ((𝑦 ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
27	26	necon1bi 2988	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅ → (𝑦 ∖ 𝐴) = ∅)
28	22, 27	syl 18	. . . . . . 7 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑦 ∖ 𝐴) = ∅)
29		ssdif0 4321	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝑦 ∖ 𝐴) = ∅)
30	28, 29	sylibr 237	. . . . . 6 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
31	30	adantlr 728	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
32		simplr 781	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑦)
33	31, 32	sseldd 3939	. . . 4 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
34	33	ex 418	. . 3 ⊢ ((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴))
35	34	exlimiv 1963	. 2 ⊢ (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴))
36	35	com12 33	1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → 𝐵 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  Tr wtr 5220

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 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-reg 9561

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4287  df-uni 4875  df-tr 5221

This theorem is used by:  setindtrs  43812