Theorem ltrelpr 7565

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 7530	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4733	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3211	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2164  ∃wrex 2473   ⊆ wss 3153  {copab 4089   × cxp 4657  ‘cfv 5254  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340  Pcnp 7351  <_P cltp 7355

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 3159  df-ss 3166  df-opab 4091  df-xp 4665  df-iltp 7530

This theorem is referenced by:  ltprordil  7649  ltexprlemm  7660  ltexprlemopl  7661  ltexprlemlol  7662  ltexprlemopu  7663  ltexprlemupu  7664  ltexprlemdisj  7666  ltexprlemloc  7667  ltexprlemfl  7669  ltexprlemrl  7670  ltexprlemfu  7671  ltexprlemru  7672  ltexpri  7673  lteupri  7677  ltaprlem  7678  prplnqu  7680  caucvgprprlemk  7743  caucvgprprlemnkltj  7749  caucvgprprlemnkeqj  7750  caucvgprprlemnjltk  7751  caucvgprprlemnbj  7753  caucvgprprlemml  7754  caucvgprprlemlol  7758  caucvgprprlemupu  7760  suplocexprlemss  7775  suplocexprlemlub  7784  gt0srpr  7808  lttrsr  7822  ltposr  7823  archsr  7842