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Theorem posi 16866
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 16863 . . 3 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
43simprbi 480 . 2 (𝐾 ∈ Poset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
5 breq1 4621 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑥))
6 breq2 4622 . . . . 5 (𝑥 = 𝑋 → (𝑋 𝑥𝑋 𝑋))
75, 6bitrd 268 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑋))
8 breq1 4621 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 breq2 4622 . . . . . 6 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
108, 9anbi12d 746 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑋 𝑦𝑦 𝑋)))
11 eqeq1 2630 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
1210, 11imbi12d 334 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ((𝑋 𝑦𝑦 𝑋) → 𝑋 = 𝑦)))
138anbi1d 740 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑋 𝑦𝑦 𝑧)))
14 breq1 4621 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
1513, 14imbi12d 334 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)))
167, 12, 153anbi123d 1396 . . 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 4622 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
18 breq1 4621 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑋𝑌 𝑋))
1917, 18anbi12d 746 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑋) ↔ (𝑋 𝑌𝑌 𝑋)))
20 eqeq2 2637 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2119, 20imbi12d 334 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑋) → 𝑋 = 𝑦) ↔ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌)))
22 breq1 4621 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
2317, 22anbi12d 746 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑧) ↔ (𝑋 𝑌𝑌 𝑧)))
2423imbi1d 331 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)))
2521, 243anbi23d 1399 . . 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 4622 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
2726anbi2d 739 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌𝑌 𝑧) ↔ (𝑋 𝑌𝑌 𝑍)))
28 breq2 4622 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2927, 28imbi12d 334 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
30293anbi3d 1402 . . 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 3314 . 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 486 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 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191   class class class wbr 4618  cfv 5850  Basecbs 15776  lecple 15864  Posetcpo 16856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-poset 16862
This theorem is referenced by:  posasymb  16868  postr  16869
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