Theorem setindtr 37591

Description: Epsilon 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 8610; 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 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Tr 𝑦
2		nfa1 2028	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)
3	1, 2	nfan 1828	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
4		eldifn 3733	. . . . . . . . . . . . . 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 4761	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
7		eldifi 3732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → 𝑥 ∈ 𝑦)
8	6, 7	impel 485	. . . . . . . . . . . . . . . . 17 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → 𝑥 ⊆ 𝑦)
9		df-ss 3588	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
10	8, 9	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
11	10	adantlr 751	. . . . . . . . . . . . . . 15 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
12	11	sseq1d 3632	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		sp 2053	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
14	13	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
15	12, 14	sylbid 230	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 → 𝑥 ∈ 𝐴))
16	5, 15	mtod 189	. . . . . . . . . . . 12 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ 𝑦) ⊆ 𝐴)
17		inssdif0 3947	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
18	16, 17	sylnib 318	. . . . . . . . . . 11 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
19	18	ex 450	. . . . . . . . . 10 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅))
20	3, 19	ralrimi 2957	. . . . . . . . 9 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
21		ralnex 2992	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
22	20, 21	sylib 208	. . . . . . . 8 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
23		vex 3203	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	23	difexi 4809	. . . . . . . . . 10 ⊢ (𝑦 ∖ 𝐴) ∈ V
25		zfreg 8500	. . . . . . . . . 10 ⊢ (((𝑦 ∖ 𝐴) ∈ V ∧ (𝑦 ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
26	24, 25	mpan 706	. . . . . . . . 9 ⊢ ((𝑦 ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
27	26	necon1bi 2822	. . . . . . . 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 3942	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝑦 ∖ 𝐴) = ∅)
30	28, 29	sylibr 224	. . . . . 6 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
31	30	adantlr 751	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
32		simplr 792	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑦)
33	31, 32	sseldd 3604	. . . 4 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
34	33	ex 450	. . 3 ⊢ ((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴))
35	34	exlimiv 1858	. 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 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437  df-tr 4753

This theorem is referenced by:  setindtrs  37592