Theorem elpotr 35961

Description: A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)

Assertion

Ref	Expression
elpotr	⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴)

Distinct variable group:   𝑧,𝐴

Proof of Theorem elpotr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alral 3066	. . . . . 6 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
2	1	alimi 1813	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑥∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
3		alral 3066	. . . . 5 ⊢ (∀𝑥∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
4	2, 3	syl 17	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
5	4	ralimi 3074	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
6		ralcom 3265	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
7		ralcom 3265	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
8	7	ralbii 3083	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9	6, 8	bitri 275	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
10	5, 9	sylib 218	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
11		dftr2 5194	. . 3 ⊢ (Tr 𝑧 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
12	11	ralbii 3083	. 2 ⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 ↔ ∀𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
13		df-po 5539	. . 3 ⊢ ( E Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
14		epel 5534	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
15		epel 5534	. . . . . . . 8 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
16	14, 15	anbi12i 629	. . . . . . 7 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
17		epel 5534	. . . . . . 7 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
18	16, 17	imbi12i 350	. . . . . 6 ⊢ (((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
19		elirrv 9512	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
20		epel 5534	. . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
21	19, 20	mtbir 323	. . . . . . 7 ⊢ ¬ 𝑥 E 𝑥
22	21	biantrur 530	. . . . . 6 ⊢ (((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
23	18, 22	bitr3i 277	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
24	23	ralbii 3083	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
25	24	2ralbii 3112	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
26	13, 25	bitr4i 278	. 2 ⊢ ( E Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
27	10, 12, 26	3imtr4i 292	1 ⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1540  ∀wral 3051   class class class wbr 5085  Tr wtr 5192   E cep 5530   Po wpo 5537

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-reg 9507

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539

This theorem is referenced by: (None)