Theorem ltrelpr 10978

Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltrelpr	⊢ <P ⊆ (P × P)

Proof of Theorem ltrelpr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltp 10965	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5753	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3983	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 400   ∈ wcel 2143   ⊆ wss 3905   ⊊ wpss 3906  {copab 5173   × cxp 5659  Pcnp 10839  <_P cltp 10843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-ss 3922  df-opab 5174  df-xp 5667  df-ltp 10965

This theorem is referenced by:  ltexpri  11023  ltaprlem  11024  ltapr  11025  suplem1pr  11032  suplem2pr  11033  supexpr  11034  ltsrpr  11057  ltsosr  11074  mappsrpr  11088