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Theorem porpss 7087
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 3855 . . . . 5 ¬ 𝑥𝑥
2 psstr 3859 . . . . 5 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
3 vex 3352 . . . . . . . 8 𝑥 ∈ V
43brrpss 7086 . . . . . . 7 (𝑥 [] 𝑥𝑥𝑥)
54notbii 309 . . . . . 6 𝑥 [] 𝑥 ↔ ¬ 𝑥𝑥)
6 vex 3352 . . . . . . . . 9 𝑦 ∈ V
76brrpss 7086 . . . . . . . 8 (𝑥 [] 𝑦𝑥𝑦)
8 vex 3352 . . . . . . . . 9 𝑧 ∈ V
98brrpss 7086 . . . . . . . 8 (𝑦 [] 𝑧𝑦𝑧)
107, 9anbi12i 604 . . . . . . 7 ((𝑥 [] 𝑦𝑦 [] 𝑧) ↔ (𝑥𝑦𝑦𝑧))
118brrpss 7086 . . . . . . 7 (𝑥 [] 𝑧𝑥𝑧)
1210, 11imbi12i 339 . . . . . 6 (((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
135, 12anbi12i 604 . . . . 5 ((¬ 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧)) ↔ (¬ 𝑥𝑥 ∧ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)))
141, 2, 13mpbir2an 682 . . . 4 𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1514rgenw 3072 . . 3 𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
1615rgen2w 3073 . 2 𝑥𝐴𝑦𝐴𝑧𝐴𝑥 [] 𝑥 ∧ ((𝑥 [] 𝑦𝑦 [] 𝑧) → 𝑥 [] 𝑧))
17 df-po 5170 . 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 382  wral 3060  wpss 3722   class class class wbr 4784   Po wpo 5168   [] crpss 7082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-po 5170  df-xp 5255  df-rel 5256  df-rpss 7083
This theorem is referenced by:  sorpss  7088  fin23lem40  9374  isfin1-3  9409  zorng  9527  fin2so  33722
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