Theorem ltrelpr 7715

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 7680	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4798	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3257	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2200  ∃wrex 2509   ⊆ wss 3198  {copab 4147   × cxp 4721  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297  Qcnq 7490  Pcnp 7501  <_P cltp 7505

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3204  df-ss 3211  df-opab 4149  df-xp 4729  df-iltp 7680

This theorem is referenced by:  ltprordil  7799  ltexprlemm  7810  ltexprlemopl  7811  ltexprlemlol  7812  ltexprlemopu  7813  ltexprlemupu  7814  ltexprlemdisj  7816  ltexprlemloc  7817  ltexprlemfl  7819  ltexprlemrl  7820  ltexprlemfu  7821  ltexprlemru  7822  ltexpri  7823  lteupri  7827  ltaprlem  7828  prplnqu  7830  caucvgprprlemk  7893  caucvgprprlemnkltj  7899  caucvgprprlemnkeqj  7900  caucvgprprlemnjltk  7901  caucvgprprlemnbj  7903  caucvgprprlemml  7904  caucvgprprlemlol  7908  caucvgprprlemupu  7910  suplocexprlemss  7925  suplocexprlemlub  7934  gt0srpr  7958  lttrsr  7972  ltposr  7973  archsr  7992