Theorem ltrelpr 7725

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 7690	. 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 6301  2^nd c2nd 6302  Qcnq 7500  Pcnp 7511  <_P cltp 7515

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 7690

This theorem is referenced by:  ltprordil  7809  ltexprlemm  7820  ltexprlemopl  7821  ltexprlemlol  7822  ltexprlemopu  7823  ltexprlemupu  7824  ltexprlemdisj  7826  ltexprlemloc  7827  ltexprlemfl  7829  ltexprlemrl  7830  ltexprlemfu  7831  ltexprlemru  7832  ltexpri  7833  lteupri  7837  ltaprlem  7838  prplnqu  7840  caucvgprprlemk  7903  caucvgprprlemnkltj  7909  caucvgprprlemnkeqj  7910  caucvgprprlemnjltk  7911  caucvgprprlemnbj  7913  caucvgprprlemml  7914  caucvgprprlemlol  7918  caucvgprprlemupu  7920  suplocexprlemss  7935  suplocexprlemlub  7944  gt0srpr  7968  lttrsr  7982  ltposr  7983  archsr  8002