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Theorem ltprord 11024
Description: Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltprord ((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))

Proof of Theorem ltprord
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2815 . . . . 5 (𝑥 = 𝐴 → (𝑥P𝐴P))
21anbi1d 629 . . . 4 (𝑥 = 𝐴 → ((𝑥P𝑦P) ↔ (𝐴P𝑦P)))
3 psseq1 4082 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
42, 3anbi12d 630 . . 3 (𝑥 = 𝐴 → (((𝑥P𝑦P) ∧ 𝑥𝑦) ↔ ((𝐴P𝑦P) ∧ 𝐴𝑦)))
5 eleq1 2815 . . . . 5 (𝑦 = 𝐵 → (𝑦P𝐵P))
65anbi2d 628 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦P) ↔ (𝐴P𝐵P)))
7 psseq2 4083 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
86, 7anbi12d 630 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦P) ∧ 𝐴𝑦) ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
9 df-ltp 10979 . . 3 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
104, 8, 9brabg 5532 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
1110bianabs 541 1 ((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wpss 3944   class class class wbr 5141  Pcnp 10853  <P cltp 10857
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-ltp 10979
This theorem is referenced by:  ltsopr  11026  ltaddpr  11028  ltexprlem7  11036  ltexpri  11037  suplem1pr  11046  suplem2pr  11047
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