Theorem porpss 7731

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 4054	. . . . 5 ⊢ ¬ 𝑥 ⊊ 𝑥
2		psstr 4059	. . . . 5 ⊢ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)
3		vex 3457	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	brrpss 7730	. . . . . . 7 ⊢ (𝑥 [⊊] 𝑥 ↔ 𝑥 ⊊ 𝑥)
5	4	notbii 323	. . . . . 6 ⊢ (¬ 𝑥 [⊊] 𝑥 ↔ ¬ 𝑥 ⊊ 𝑥)
6		vex 3457	. . . . . . . . 9 ⊢ 𝑦 ∈ V
7	6	brrpss 7730	. . . . . . . 8 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
8		vex 3457	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	brrpss 7730	. . . . . . . 8 ⊢ (𝑦 [⊊] 𝑧 ↔ 𝑦 ⊊ 𝑧)
10	7, 9	anbi12i 640	. . . . . . 7 ⊢ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) ↔ (𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧))
11	8	brrpss 7730	. . . . . . 7 ⊢ (𝑥 [⊊] 𝑧 ↔ 𝑥 ⊊ 𝑧)
12	10, 11	imbi12i 353	. . . . . 6 ⊢ (((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧) ↔ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧))
13	5, 12	anbi12i 640	. . . . 5 ⊢ ((¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) ↔ (¬ 𝑥 ⊊ 𝑥 ∧ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)))
14	1, 2, 13	mpbir2an 724	. . . 4 ⊢ (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
15	14	rgenw 3082	. . 3 ⊢ ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
16	15	rgen2w 3083	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))
17		df-po 5567	. 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

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wral 3078   ⊊ wpss 3903   class class class wbr 5107   Po wpo 5565   [⊊] crpss 7726

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-po 5567  df-xp 5665  df-rel 5666  df-rpss 7727

This theorem is used by:  sorpss  7732  fin23lem40  10356  isfin1-3  10391  zorng  10509  fin2so  38348