Theorem ltrelpr 7306

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 7271	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4608	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3124	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 103   ∈ wcel 1480  ∃wrex 2415   ⊆ wss 3066  {copab 3983   × cxp 4532  ‘cfv 5118  1^st c1st 6029  2^nd c2nd 6030  Qcnq 7081  Pcnp 7092  <_P cltp 7096

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-opab 3985  df-xp 4540  df-iltp 7271

This theorem is referenced by:  ltprordil  7390  ltexprlemm  7401  ltexprlemopl  7402  ltexprlemlol  7403  ltexprlemopu  7404  ltexprlemupu  7405  ltexprlemdisj  7407  ltexprlemloc  7408  ltexprlemfl  7410  ltexprlemrl  7411  ltexprlemfu  7412  ltexprlemru  7413  ltexpri  7414  lteupri  7418  ltaprlem  7419  prplnqu  7421  caucvgprprlemk  7484  caucvgprprlemnkltj  7490  caucvgprprlemnkeqj  7491  caucvgprprlemnjltk  7492  caucvgprprlemnbj  7494  caucvgprprlemml  7495  caucvgprprlemlol  7499  caucvgprprlemupu  7501  suplocexprlemss  7516  suplocexprlemlub  7525  gt0srpr  7549  lttrsr  7563  ltposr  7564  archsr  7583