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Theorem porpss 7580
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.
StepHypRef Expression
1 pssirr 4035 . . . . 5 ¬ 𝑥𝑥
2 psstr 4039 . . . . 5 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
3 vex 3436 . . . . . . . 8 𝑥 ∈ V
43brrpss 7579 . . . . . . 7 (𝑥 [] 𝑥𝑥𝑥)
54notbii 320 . . . . . 6 𝑥 [] 𝑥 ↔ ¬ 𝑥𝑥)
6 vex 3436 . . . . . . . . 9 𝑦 ∈ V
76brrpss 7579 . . . . . . . 8 (𝑥 [] 𝑦𝑥𝑦)
8 vex 3436 . . . . . . . . 9 𝑧 ∈ V
98brrpss 7579 . . . . . . . 8 (𝑦 [] 𝑧𝑦𝑧)
107, 9anbi12i 627 . . . . . . 7 ((𝑥 [] 𝑦𝑦 [] 𝑧) ↔ (𝑥𝑦𝑦𝑧))
118brrpss 7579 . . . . . . 7 (𝑥 [] 𝑧𝑥𝑧)
1210, 11imbi12i 351 . . . . . 6 (((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
135, 12anbi12i 627 . . . . 5 ((¬ 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)) ↔ (¬ 𝑥𝑥 ∧ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)))
141, 2, 13mpbir2an 708 . . . 4 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1514rgenw 3076 . . 3 𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1615rgen2w 3077 . 2 𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
17 df-po 5503 . 2 ( [] Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)))
1816, 17mpbir 230 1 [] Po 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wral 3064  wpss 3888   class class class wbr 5074   Po wpo 5501   [] crpss 7575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-po 5503  df-xp 5595  df-rel 5596  df-rpss 7576
This theorem is referenced by:  sorpss  7581  fin23lem40  10107  isfin1-3  10142  zorng  10260  fin2so  35764
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