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Theorem porpss 7026
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 3782 . . . . 5 ¬ 𝑥𝑥
2 psstr 3786 . . . . 5 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
3 vex 3275 . . . . . . . 8 𝑥 ∈ V
43brrpss 7025 . . . . . . 7 (𝑥 [] 𝑥𝑥𝑥)
54notbii 309 . . . . . 6 𝑥 [] 𝑥 ↔ ¬ 𝑥𝑥)
6 vex 3275 . . . . . . . . 9 𝑦 ∈ V
76brrpss 7025 . . . . . . . 8 (𝑥 [] 𝑦𝑥𝑦)
8 vex 3275 . . . . . . . . 9 𝑧 ∈ V
98brrpss 7025 . . . . . . . 8 (𝑦 [] 𝑧𝑦𝑧)
107, 9anbi12i 735 . . . . . . 7 ((𝑥 [] 𝑦𝑦 [] 𝑧) ↔ (𝑥𝑦𝑦𝑧))
118brrpss 7025 . . . . . . 7 (𝑥 [] 𝑧𝑥𝑧)
1210, 11imbi12i 339 . . . . . 6 (((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
135, 12anbi12i 735 . . . . 5 ((¬ 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)) ↔ (¬ 𝑥𝑥 ∧ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)))
141, 2, 13mpbir2an 993 . . . 4 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1514rgenw 2994 . . 3 𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1615rgen2w 2995 . 2 𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
17 df-po 5107 . 2 ( [] Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)))
1816, 17mpbir 221 1 [] Po 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wral 2982  wpss 3649   class class class wbr 4728   Po wpo 5105   [] crpss 7021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pr 4979
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-sn 4254  df-pr 4256  df-op 4260  df-br 4729  df-opab 4789  df-po 5107  df-xp 5192  df-rel 5193  df-rpss 7022
This theorem is referenced by:  sorpss  7027  fin23lem40  9254  isfin1-3  9289  zorng  9407  fin2so  33596
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