Theorem ltrelpr 7878

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

Assertion

Ref	Expression
ltrelpr	⊢ <P ⊆ (P × P)

Proof of Theorem ltrelpr

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

Step	Hyp	Ref	Expression
1		df-iltp 7843	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4851	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3280	1 ⊢ <P ⊆ (P × P)

This proof depends on syntax axioms:   ∧ wa 104   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  {copab 4192   × cxp 4774  ‘cfv 5380  1^st c1st 6376  2^nd c2nd 6377  Qcnq 7653  Pcnp 7664  <_P cltp 7668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4194  df-xp 4782  df-iltp 7843

This theorem is used by:  ltprordil  7962  ltexprlemm  7973  ltexprlemopl  7974  ltexprlemlol  7975  ltexprlemopu  7976  ltexprlemupu  7977  ltexprlemdisj  7979  ltexprlemloc  7980  ltexprlemfl  7982  ltexprlemrl  7983  ltexprlemfu  7984  ltexprlemru  7985  ltexpri  7986  lteupri  7990  ltaprlem  7991  prplnqu  7993  caucvgprprlemk  8056  caucvgprprlemnkltj  8062  caucvgprprlemnkeqj  8063  caucvgprprlemnjltk  8064  caucvgprprlemnbj  8066  caucvgprprlemml  8067  caucvgprprlemlol  8071  caucvgprprlemupu  8073  suplocexprlemss  8088  suplocexprlemlub  8097  gt0srpr  8121  lttrsr  8135  ltposr  8136  archsr  8155