Theorem truniALTVD 38136

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 37772 is truniALTVD 38136 without virtual deductions and was automatically derived from truniALTVD 38136.

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 37859	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   )
2		simpr 476	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑦 ∈ ∪ 𝐴)
3	1, 2	e2 37877	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑦 ∈ ∪ 𝐴   )
4		eluni 4375	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴))
5	4	biimpi 205	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 → ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴))
6	3, 5	e2 37877	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
7		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ 𝑦)
8	1, 7	e2 37877	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ 𝑦   )
9		idn3 37861	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   )
10		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑦 ∈ 𝑞)
11	9, 10	e3 37985	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑦 ∈ 𝑞   )
12		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
13	9, 12	e3 37985	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑞 ∈ 𝐴   )
14		idn1 37811	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
15		rspsbc 3484	. . . . . . . . . . . . . 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 37996	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   [𝑞 / 𝑥]Tr 𝑥   )
18		trsbc 37771	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
19	18	biimpd 218	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
20	13, 17, 19	e33 37982	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   Tr 𝑞   )
21		trel 4687	. . . . . . . . . . . 12 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
22	21	expdcom 454	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → (Tr 𝑞 → 𝑧 ∈ 𝑞)))
23	8, 11, 20, 22	e233 38013	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ 𝑞   )
24		elunii 4377	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
25	24	ex 449	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴))
26	23, 13, 25	e33 37982	. . . . . . . . 9 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ,   (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
27	26	in3 37855	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
28	27	gen21 37865	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
29		19.23v 1889	. . . . . . . 8 ⊢ (∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) ↔ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴))
30	29	biimpi 205	. . . . . . 7 ⊢ (∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) → (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴))
31	28, 30	e2 37877	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
32		pm2.27 41	. . . . . 6 ⊢ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → ((∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
33	6, 31, 32	e22 37917	. . . . 5 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴)   ▶   𝑧 ∈ ∪ 𝐴   )
34	33	in2 37851	. . . 4 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
35	34	gen12 37864	. . 3 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴)   )
36		dftr2 4682	. . . 4 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
37	36	biimpri 217	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) → Tr ∪ 𝐴)
38	35, 37	e1a 37873	. 2 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   Tr ∪ 𝐴   )
39	38	in1 37808	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  [wsbc 3402  ∪ cuni 4372  Tr wtr 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-sbc 3403  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681  df-vd1 37807  df-vd2 37815  df-vd3 37827

This theorem is referenced by: (None)