Theorem ltrelpr 7724

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 7689	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4800	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3259	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2202  ∃wrex 2511   ⊆ wss 3200  {copab 4149   × cxp 4723  ‘cfv 5326  1^st c1st 6300  2^nd c2nd 6301  Qcnq 7499  Pcnp 7510  <_P cltp 7514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-opab 4151  df-xp 4731  df-iltp 7689

This theorem is referenced by:  ltprordil  7808  ltexprlemm  7819  ltexprlemopl  7820  ltexprlemlol  7821  ltexprlemopu  7822  ltexprlemupu  7823  ltexprlemdisj  7825  ltexprlemloc  7826  ltexprlemfl  7828  ltexprlemrl  7829  ltexprlemfu  7830  ltexprlemru  7831  ltexpri  7832  lteupri  7836  ltaprlem  7837  prplnqu  7839  caucvgprprlemk  7902  caucvgprprlemnkltj  7908  caucvgprprlemnkeqj  7909  caucvgprprlemnjltk  7910  caucvgprprlemnbj  7912  caucvgprprlemml  7913  caucvgprprlemlol  7917  caucvgprprlemupu  7919  suplocexprlemss  7934  suplocexprlemlub  7943  gt0srpr  7967  lttrsr  7981  ltposr  7982  archsr  8001