Theorem ltrelpr 11096

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 11083	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5748	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3959	1 ⊢ <P ⊆ (P × P)

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2146   ⊆ wss 3900   ⊊ wpss 4016  {copab 5170   × cxp 5653  Pcnp 10957  <_P cltp 10961

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-ss 3901  df-opab 5171  df-xp 5661  df-ltp 11083

This theorem is used by:  ltexpri  11141  ltaprlem  11142  ltapr  11143  suplem1pr  11150  suplem2pr  11151  supexpr  11152  ltsrpr  11175  ltsosr  11192  mappsrpr  11206