Theorem ltrelpr 6834

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 6799	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4461	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3039	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 102   ∈ wcel 1434  ∃wrex 2354   ⊆ wss 2983  {copab 3859   × cxp 4390  ‘cfv 4953  1^st c1st 5818  2^nd c2nd 5819  Qcnq 6609  Pcnp 6620  <_P cltp 6624

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2989  df-ss 2996  df-opab 3861  df-xp 4398  df-iltp 6799

This theorem is referenced by:  ltprordil  6918  ltexprlemm  6929  ltexprlemopl  6930  ltexprlemlol  6931  ltexprlemopu  6932  ltexprlemupu  6933  ltexprlemdisj  6935  ltexprlemloc  6936  ltexprlemfl  6938  ltexprlemrl  6939  ltexprlemfu  6940  ltexprlemru  6941  ltexpri  6942  lteupri  6946  ltaprlem  6947  prplnqu  6949  caucvgprprlemk  7012  caucvgprprlemnkltj  7018  caucvgprprlemnkeqj  7019  caucvgprprlemnjltk  7020  caucvgprprlemnbj  7022  caucvgprprlemml  7023  caucvgprprlemlol  7027  caucvgprprlemupu  7029  gt0srpr  7064  lttrsr  7078  ltposr  7079  archsr  7097