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Theorem porpss 7746
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 4113 . . . . 5 ¬ 𝑥𝑥
2 psstr 4117 . . . . 5 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
3 vex 3482 . . . . . . . 8 𝑥 ∈ V
43brrpss 7745 . . . . . . 7 (𝑥 [] 𝑥𝑥𝑥)
54notbii 320 . . . . . 6 𝑥 [] 𝑥 ↔ ¬ 𝑥𝑥)
6 vex 3482 . . . . . . . . 9 𝑦 ∈ V
76brrpss 7745 . . . . . . . 8 (𝑥 [] 𝑦𝑥𝑦)
8 vex 3482 . . . . . . . . 9 𝑧 ∈ V
98brrpss 7745 . . . . . . . 8 (𝑦 [] 𝑧𝑦𝑧)
107, 9anbi12i 628 . . . . . . 7 ((𝑥 [] 𝑦𝑦 [] 𝑧) ↔ (𝑥𝑦𝑦𝑧))
118brrpss 7745 . . . . . . 7 (𝑥 [] 𝑧𝑥𝑧)
1210, 11imbi12i 350 . . . . . 6 (((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
135, 12anbi12i 628 . . . . 5 ((¬ 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)) ↔ (¬ 𝑥𝑥 ∧ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)))
141, 2, 13mpbir2an 711 . . . 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 5597 . 2 ( [] Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)))
1816, 17mpbir 231 1 [] Po 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wral 3059  wpss 3964   class class class wbr 5148   Po wpo 5595   [] crpss 7741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-po 5597  df-xp 5695  df-rel 5696  df-rpss 7742
This theorem is referenced by:  sorpss  7747  fin23lem40  10389  isfin1-3  10424  zorng  10542  fin2so  37594
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