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Theorem posi 17552
Description: Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
posi.b 𝐵 = (Base‘𝐾)
posi.l = (le‘𝐾)
Assertion
Ref Expression
posi ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Proof of Theorem posi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 posi.b . . . 4 𝐵 = (Base‘𝐾)
2 posi.l . . . 4 = (le‘𝐾)
31, 2ispos 17549 . . 3 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
43simprbi 499 . 2 (𝐾 ∈ Poset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
5 breq1 5060 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑥))
6 breq2 5061 . . . . 5 (𝑥 = 𝑋 → (𝑋 𝑥𝑋 𝑋))
75, 6bitrd 281 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑋))
8 breq1 5060 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 breq2 5061 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
108, 9anbi12d 632 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑋 𝑦𝑦 𝑋)))
11 eqeq1 2823 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
1210, 11imbi12d 347 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ((𝑋 𝑦𝑦 𝑋) → 𝑋 = 𝑦)))
138anbi1d 631 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑋 𝑦𝑦 𝑧)))
14 breq1 5060 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
1513, 14imbi12d 347 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)))
167, 12, 153anbi123d 1429 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑋) → 𝑋 = 𝑦) ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧))))
17 breq2 5061 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
18 breq1 5060 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑋𝑌 𝑋))
1917, 18anbi12d 632 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑋) ↔ (𝑋 𝑌𝑌 𝑋)))
20 eqeq2 2831 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2119, 20imbi12d 347 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑋) → 𝑋 = 𝑦) ↔ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌)))
22 breq1 5060 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
2317, 22anbi12d 632 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑧) ↔ (𝑋 𝑌𝑌 𝑧)))
2423imbi1d 344 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)))
2521, 243anbi23d 1432 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑋) → 𝑋 = 𝑦) ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧))))
26 breq2 5061 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
2726anbi2d 630 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌𝑌 𝑧) ↔ (𝑋 𝑌𝑌 𝑍)))
28 breq2 5061 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2927, 28imbi12d 347 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
30293anbi3d 1435 . . 3 (𝑧 = 𝑍 → ((𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
3116, 25, 30rspc3v 3634 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
324, 31mpan9 509 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493   class class class wbr 5057  cfv 6348  Basecbs 16475  lecple 16564  Posetcpo 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-poset 17548
This theorem is referenced by:  posasymb  17554  postr  17555
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