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Theorem porpss 7721
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 4101 . . . . 5 ¬ 𝑥𝑥
2 psstr 4105 . . . . 5 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
3 vex 3476 . . . . . . . 8 𝑥 ∈ V
43brrpss 7720 . . . . . . 7 (𝑥 [] 𝑥𝑥𝑥)
54notbii 319 . . . . . 6 𝑥 [] 𝑥 ↔ ¬ 𝑥𝑥)
6 vex 3476 . . . . . . . . 9 𝑦 ∈ V
76brrpss 7720 . . . . . . . 8 (𝑥 [] 𝑦𝑥𝑦)
8 vex 3476 . . . . . . . . 9 𝑧 ∈ V
98brrpss 7720 . . . . . . . 8 (𝑦 [] 𝑧𝑦𝑧)
107, 9anbi12i 625 . . . . . . 7 ((𝑥 [] 𝑦𝑦 [] 𝑧) ↔ (𝑥𝑦𝑦𝑧))
118brrpss 7720 . . . . . . 7 (𝑥 [] 𝑧𝑥𝑧)
1210, 11imbi12i 349 . . . . . 6 (((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
135, 12anbi12i 625 . . . . 5 ((¬ 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)) ↔ (¬ 𝑥𝑥 ∧ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)))
141, 2, 13mpbir2an 707 . . . 4 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1514rgenw 3063 . . 3 𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1615rgen2w 3064 . 2 𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
17 df-po 5589 . 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 394  wral 3059  wpss 3950   class class class wbr 5149   Po wpo 5587   [] crpss 7716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-xp 5683  df-rel 5684  df-rpss 7717
This theorem is referenced by:  sorpss  7722  fin23lem40  10350  isfin1-3  10385  zorng  10503  fin2so  36780
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