Theorem setindtr 43020

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 9694; 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 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Tr 𝑦
2		nfa1 2152	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)
3	1, 2	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
4		eldifn 4098	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
5	4	adantl 481	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
6		trss 5228	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
7		eldifi 4097	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑦 ∖ 𝐴) → 𝑥 ∈ 𝑦)
8	6, 7	impel 505	. . . . . . . . . . . . . . . . 17 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → 𝑥 ⊆ 𝑦)
9		dfss2 3935	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
10	8, 9	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
11	10	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ∩ 𝑦) = 𝑥)
12	11	sseq1d 3981	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
13		sp 2184	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
14	13	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
15	12, 14	sylbid 240	. . . . . . . . . . . . 13 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ((𝑥 ∩ 𝑦) ⊆ 𝐴 → 𝑥 ∈ 𝐴))
16	5, 15	mtod 198	. . . . . . . . . . . 12 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ 𝑦) ⊆ 𝐴)
17		inssdif0 4340	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
18	16, 17	sylnib 328	. . . . . . . . . . 11 ⊢ (((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (𝑦 ∖ 𝐴)) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
19	18	ex 412	. . . . . . . . . 10 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (𝑦 ∖ 𝐴) → ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅))
20	3, 19	ralrimi 3236	. . . . . . . . 9 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
21		ralnex 3056	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑦 ∖ 𝐴) ¬ (𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
22	20, 21	sylib 218	. . . . . . . 8 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ¬ ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
23		vex 3454	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	23	difexi 5288	. . . . . . . . . 10 ⊢ (𝑦 ∖ 𝐴) ∈ V
25		zfreg 9555	. . . . . . . . . 10 ⊢ (((𝑦 ∖ 𝐴) ∈ V ∧ (𝑦 ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
26	24, 25	mpan 690	. . . . . . . . 9 ⊢ ((𝑦 ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦 ∖ 𝐴)(𝑥 ∩ (𝑦 ∖ 𝐴)) = ∅)
27	26	necon1bi 2954	. . . . . . . 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 4332	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝑦 ∖ 𝐴) = ∅)
30	28, 29	sylibr 234	. . . . . 6 ⊢ ((Tr 𝑦 ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
31	30	adantlr 715	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝑦 ⊆ 𝐴)
32		simplr 768	. . . . 5 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑦)
33	31, 32	sseldd 3950	. . . 4 ⊢ (((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) ∧ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
34	33	ex 412	. . 3 ⊢ ((Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴))
35	34	exlimiv 1930	. 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 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  Tr wtr 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-reg 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300  df-uni 4875  df-tr 5218

This theorem is referenced by:  setindtrs  43021