Theorem ltrelpr 7446

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 7411	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4678	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3174	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 103   ∈ wcel 2136  ∃wrex 2445   ⊆ wss 3116  {copab 4042   × cxp 4602  ‘cfv 5188  1^st c1st 6106  2^nd c2nd 6107  Qcnq 7221  Pcnp 7232  <_P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-opab 4044  df-xp 4610  df-iltp 7411

This theorem is referenced by:  ltprordil  7530  ltexprlemm  7541  ltexprlemopl  7542  ltexprlemlol  7543  ltexprlemopu  7544  ltexprlemupu  7545  ltexprlemdisj  7547  ltexprlemloc  7548  ltexprlemfl  7550  ltexprlemrl  7551  ltexprlemfu  7552  ltexprlemru  7553  ltexpri  7554  lteupri  7558  ltaprlem  7559  prplnqu  7561  caucvgprprlemk  7624  caucvgprprlemnkltj  7630  caucvgprprlemnkeqj  7631  caucvgprprlemnjltk  7632  caucvgprprlemnbj  7634  caucvgprprlemml  7635  caucvgprprlemlol  7639  caucvgprprlemupu  7641  suplocexprlemss  7656  suplocexprlemlub  7665  gt0srpr  7689  lttrsr  7703  ltposr  7704  archsr  7723