Theorem ltrelpr 7567

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 7532	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4734	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3212	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2164  ∃wrex 2473   ⊆ wss 3154  {copab 4090   × cxp 4658  ‘cfv 5255  1^st c1st 6193  2^nd c2nd 6194  Qcnq 7342  Pcnp 7353  <_P cltp 7357

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3160  df-ss 3167  df-opab 4092  df-xp 4666  df-iltp 7532

This theorem is referenced by:  ltprordil  7651  ltexprlemm  7662  ltexprlemopl  7663  ltexprlemlol  7664  ltexprlemopu  7665  ltexprlemupu  7666  ltexprlemdisj  7668  ltexprlemloc  7669  ltexprlemfl  7671  ltexprlemrl  7672  ltexprlemfu  7673  ltexprlemru  7674  ltexpri  7675  lteupri  7679  ltaprlem  7680  prplnqu  7682  caucvgprprlemk  7745  caucvgprprlemnkltj  7751  caucvgprprlemnkeqj  7752  caucvgprprlemnjltk  7753  caucvgprprlemnbj  7755  caucvgprprlemml  7756  caucvgprprlemlol  7760  caucvgprprlemupu  7762  suplocexprlemss  7777  suplocexprlemlub  7786  gt0srpr  7810  lttrsr  7824  ltposr  7825  archsr  7844