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Theorem ltprord 11070
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 2829 . . . . 5 (𝑥 = 𝐴 → (𝑥P𝐴P))
21anbi1d 631 . . . 4 (𝑥 = 𝐴 → ((𝑥P𝑦P) ↔ (𝐴P𝑦P)))
3 psseq1 4090 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
42, 3anbi12d 632 . . 3 (𝑥 = 𝐴 → (((𝑥P𝑦P) ∧ 𝑥𝑦) ↔ ((𝐴P𝑦P) ∧ 𝐴𝑦)))
5 eleq1 2829 . . . . 5 (𝑦 = 𝐵 → (𝑦P𝐵P))
65anbi2d 630 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦P) ↔ (𝐴P𝐵P)))
7 psseq2 4091 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
86, 7anbi12d 632 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦P) ∧ 𝐴𝑦) ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
9 df-ltp 11025 . . 3 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
104, 8, 9brabg 5544 . 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 206  wa 395   = wceq 1540  wcel 2108  wpss 3952   class class class wbr 5143  Pcnp 10899  <P cltp 10903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-ltp 11025
This theorem is referenced by:  ltsopr  11072  ltaddpr  11074  ltexprlem7  11082  ltexpri  11083  suplem1pr  11092  suplem2pr  11093
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