Theorem porpss 7674

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 4037	. . . . 5 ⊢ ¬ 𝑥 ⊊ 𝑥
2		psstr 4041	. . . . 5 ⊢ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)
3		vex 3437	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	brrpss 7673	. . . . . . 7 ⊢ (𝑥 [⊊] 𝑥 ↔ 𝑥 ⊊ 𝑥)
5	4	notbii 322	. . . . . 6 ⊢ (¬ 𝑥 [⊊] 𝑥 ↔ ¬ 𝑥 ⊊ 𝑥)
6		vex 3437	. . . . . . . . 9 ⊢ 𝑦 ∈ V
7	6	brrpss 7673	. . . . . . . 8 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
8		vex 3437	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	brrpss 7673	. . . . . . . 8 ⊢ (𝑦 [⊊] 𝑧 ↔ 𝑦 ⊊ 𝑧)
10	7, 9	anbi12i 635	. . . . . . 7 ⊢ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) ↔ (𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧))
11	8	brrpss 7673	. . . . . . 7 ⊢ (𝑥 [⊊] 𝑧 ↔ 𝑥 ⊊ 𝑧)
12	10, 11	imbi12i 352	. . . . . 6 ⊢ (((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧) ↔ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧))
13	5, 12	anbi12i 635	. . . . 5 ⊢ ((¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) ↔ (¬ 𝑥 ⊊ 𝑥 ∧ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)))
14	1, 2, 13	mpbir2an 718	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
15	14	rgenw 3059	. . 3 ⊢ ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
16	15	rgen2w 3060	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
17		df-po 5529	. 2 ⊢ ( [⊊] Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)))
18	16, 17	mpbir 233	1 ⊢ [⊊] Po 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397  ∀wral 3055   ⊊ wpss 3886   class class class wbr 5075   Po wpo 5527   [⊊] crpss 7669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-po 5529  df-xp 5627  df-rel 5628  df-rpss 7670

This theorem is referenced by:  sorpss  7675  fin23lem40  10268  isfin1-3  10303  zorng  10421  fin2so  37989