Theorem truniALTVD 41584

Description: The union of a class of transitive sets is transitive. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. truniALT 41247 is truniALTVD 41584 without virtual deductions and was automatically derived from truniALTVD 41584.

1::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
2::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   )
3:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑦 ∈ ∪ 𝐴   )
5:4:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
6::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
7:6:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑦 ∈ 𝑞   )
8:6:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑞 ∈ 𝐴   )
9:1,8:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:8,9:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   Tr 𝑞   )
11:3,7,10:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ 𝑞   )
12:11,8:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
13:12:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
14:13:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
15:14:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
16:5,15:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
17:16:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
18:17:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥    ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
19:18:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∪ 𝐴   )
qed:19:	⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∪ 𝐴)

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
truniALTVD	⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem truniALTVD

Dummy variables 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn2 41319	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   )
2		simpr 488	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑦 ∈ ∪ 𝐴)
3	1, 2	e2 41337	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑦 ∈ ∪ 𝐴   )
4		eluni 4803	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴))
5	4	biimpi 219	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 → ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴))
6	3, 5	e2 41337	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
7		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ 𝑦)
8	1, 7	e2 41337	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ 𝑦   )
9		idn3 41321	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
10		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑦 ∈ 𝑞)
11	9, 10	e3 41443	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑦 ∈ 𝑞   )
12		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
13	9, 12	e3 41443	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑞 ∈ 𝐴   )
14		idn1 41280	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
15		rspsbc 3808	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
16	15	com12 32	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥))
17	14, 13, 16	e13 41454	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
18		trsbc 41246	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
19	18	biimpd 232	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
20	13, 17, 19	e33 41440	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   Tr 𝑞   )
21		trel 5143	. . . . . . . . . . . 12 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
22	21	expdcom 418	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → (Tr 𝑞 → 𝑧 ∈ 𝑞)))
23	8, 11, 20, 22	e233 41471	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ 𝑞   )
24		elunii 4805	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
25	24	ex 416	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴))
26	23, 13, 25	e33 41440	. . . . . . . . 9 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
27	26	in3 41315	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
28	27	gen21 41325	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
29		19.23v 1943	. . . . . . . 8 ⊢ (∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) ↔ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴))
30	29	biimpi 219	. . . . . . 7 ⊢ (∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) → (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴))
31	28, 30	e2 41337	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
32		pm2.27 42	. . . . . 6 ⊢ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → ((∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
33	6, 31, 32	e22 41377	. . . . 5 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
34	33	in2 41311	. . . 4 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
35	34	gen12 41324	. . 3 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
36		dftr2 5138	. . . 4 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
37	36	biimpri 231	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) → Tr ∪ 𝐴)
38	35, 37	e1a 41333	. 2 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   Tr ∪ 𝐴   )
39	38	in1 41277	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  [wsbc 3720  ∪ cuni 4800  Tr wtr 5136

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-sbc 3721  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-vd1 41276  df-vd2 41284  df-vd3 41296

This theorem is referenced by: (None)