Theorem porpss 7472

Description: Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
porpss	⊢ [⊊] Po 𝐴

Proof of Theorem porpss

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

Step	Hyp	Ref	Expression
1		pssirr 3992	. . . . 5 ⊢ ¬ 𝑥 ⊊ 𝑥
2		psstr 3996	. . . . 5 ⊢ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)
3		vex 3402	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	brrpss 7471	. . . . . . 7 ⊢ (𝑥 [⊊] 𝑥 ↔ 𝑥 ⊊ 𝑥)
5	4	notbii 323	. . . . . 6 ⊢ (¬ 𝑥 [⊊] 𝑥 ↔ ¬ 𝑥 ⊊ 𝑥)
6		vex 3402	. . . . . . . . 9 ⊢ 𝑦 ∈ V
7	6	brrpss 7471	. . . . . . . 8 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
8		vex 3402	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	brrpss 7471	. . . . . . . 8 ⊢ (𝑦 [⊊] 𝑧 ↔ 𝑦 ⊊ 𝑧)
10	7, 9	anbi12i 630	. . . . . . 7 ⊢ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) ↔ (𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧))
11	8	brrpss 7471	. . . . . . 7 ⊢ (𝑥 [⊊] 𝑧 ↔ 𝑥 ⊊ 𝑧)
12	10, 11	imbi12i 354	. . . . . 6 ⊢ (((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧) ↔ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧))
13	5, 12	anbi12i 630	. . . . 5 ⊢ ((¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) ↔ (¬ 𝑥 ⊊ 𝑥 ∧ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)))
14	1, 2, 13	mpbir2an 711	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
15	14	rgenw 3065	. . 3 ⊢ ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
16	15	rgen2w 3066	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
17		df-po 5443	. 2 ⊢ ( [⊊] Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)))
18	16, 17	mpbir 234	1 ⊢ [⊊] Po 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wral 3053   ⊊ wpss 3845   class class class wbr 5031   Po wpo 5441   [⊊] crpss 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-po 5443  df-xp 5532  df-rel 5533  df-rpss 7468

This theorem is referenced by:  sorpss  7473  fin23lem40  9852  isfin1-3  9887  zorng  10005  fin2so  35384