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Theorem ltprord 10440
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 2897 . . . . 5 (𝑥 = 𝐴 → (𝑥P𝐴P))
21anbi1d 629 . . . 4 (𝑥 = 𝐴 → ((𝑥P𝑦P) ↔ (𝐴P𝑦P)))
3 psseq1 4061 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
42, 3anbi12d 630 . . 3 (𝑥 = 𝐴 → (((𝑥P𝑦P) ∧ 𝑥𝑦) ↔ ((𝐴P𝑦P) ∧ 𝐴𝑦)))
5 eleq1 2897 . . . . 5 (𝑦 = 𝐵 → (𝑦P𝐵P))
65anbi2d 628 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦P) ↔ (𝐴P𝐵P)))
7 psseq2 4062 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
86, 7anbi12d 630 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦P) ∧ 𝐴𝑦) ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
9 df-ltp 10395 . . 3 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
104, 8, 9brabg 5417 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ((𝐴P𝐵P) ∧ 𝐴𝐵)))
1110bianabs 542 1 ((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wpss 3934   class class class wbr 5057  Pcnp 10269  <P cltp 10273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-ltp 10395
This theorem is referenced by:  ltsopr  10442  ltaddpr  10444  ltexprlem7  10452  ltexpri  10453  suplem1pr  10462  suplem2pr  10463
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