Theorem ltrelpr 7837

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 7802	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4830	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3274	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2205  ∃wrex 2523   ⊆ wss 3214  {copab 4176   × cxp 4753  ‘cfv 5358  1^st c1st 6346  2^nd c2nd 6347  Qcnq 7612  Pcnp 7623  <_P cltp 7627

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3220  df-ss 3227  df-opab 4178  df-xp 4761  df-iltp 7802

This theorem is referenced by:  ltprordil  7921  ltexprlemm  7932  ltexprlemopl  7933  ltexprlemlol  7934  ltexprlemopu  7935  ltexprlemupu  7936  ltexprlemdisj  7938  ltexprlemloc  7939  ltexprlemfl  7941  ltexprlemrl  7942  ltexprlemfu  7943  ltexprlemru  7944  ltexpri  7945  lteupri  7949  ltaprlem  7950  prplnqu  7952  caucvgprprlemk  8015  caucvgprprlemnkltj  8021  caucvgprprlemnkeqj  8022  caucvgprprlemnjltk  8023  caucvgprprlemnbj  8025  caucvgprprlemml  8026  caucvgprprlemlol  8030  caucvgprprlemupu  8032  suplocexprlemss  8047  suplocexprlemlub  8056  gt0srpr  8080  lttrsr  8094  ltposr  8095  archsr  8114