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Theorem ltprord 10530
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 2820 . . . . 5 (𝑥 = 𝐴 → (𝑥P𝐴P))
21anbi1d 633 . . . 4 (𝑥 = 𝐴 → ((𝑥P𝑦P) ↔ (𝐴P𝑦P)))
3 psseq1 3978 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
42, 3anbi12d 634 . . 3 (𝑥 = 𝐴 → (((𝑥P𝑦P) ∧ 𝑥𝑦) ↔ ((𝐴P𝑦P) ∧ 𝐴𝑦)))
5 eleq1 2820 . . . . 5 (𝑦 = 𝐵 → (𝑦P𝐵P))
65anbi2d 632 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦P) ↔ (𝐴P𝐵P)))
7 psseq2 3979 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
86, 7anbi12d 634 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦P) ∧ 𝐴𝑦) ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
9 df-ltp 10485 . . 3 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
104, 8, 9brabg 5394 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
1110bianabs 545 1 ((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wpss 3844   class class class wbr 5030  Pcnp 10359  <P cltp 10363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-ltp 10485
This theorem is referenced by:  ltsopr  10532  ltaddpr  10534  ltexprlem7  10542  ltexpri  10543  suplem1pr  10552  suplem2pr  10553
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