Theorem ltrelpr 7768

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 7733	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4806	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3260	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2202  ∃wrex 2512   ⊆ wss 3201  {copab 4154   × cxp 4729  ‘cfv 5333  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7543  Pcnp 7554  <_P cltp 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214  df-opab 4156  df-xp 4737  df-iltp 7733

This theorem is referenced by:  ltprordil  7852  ltexprlemm  7863  ltexprlemopl  7864  ltexprlemlol  7865  ltexprlemopu  7866  ltexprlemupu  7867  ltexprlemdisj  7869  ltexprlemloc  7870  ltexprlemfl  7872  ltexprlemrl  7873  ltexprlemfu  7874  ltexprlemru  7875  ltexpri  7876  lteupri  7880  ltaprlem  7881  prplnqu  7883  caucvgprprlemk  7946  caucvgprprlemnkltj  7952  caucvgprprlemnkeqj  7953  caucvgprprlemnjltk  7954  caucvgprprlemnbj  7956  caucvgprprlemml  7957  caucvgprprlemlol  7961  caucvgprprlemupu  7963  suplocexprlemss  7978  suplocexprlemlub  7987  gt0srpr  8011  lttrsr  8025  ltposr  8026  archsr  8045