Theorem ltrelpr 11083

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 11070	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
2		opabssxp 5743	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P)
3	1, 2	eqsstri 3977	1 ⊢ <P ⊆ (P × P)

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2145   ⊆ wss 3899   ⊊ wpss 3900  {copab 5167   × cxp 5649  Pcnp 10944  <_P cltp 10948

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-ss 3916  df-opab 5168  df-xp 5657  df-ltp 11070

This theorem is used by:  ltexpri  11128  ltaprlem  11129  ltapr  11130  suplem1pr  11137  suplem2pr  11138  supexpr  11139  ltsrpr  11162  ltsosr  11179  mappsrpr  11193