Theorem ltrelpr 7688

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 7653	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4792	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3256	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2200  ∃wrex 2509   ⊆ wss 3197  {copab 4143   × cxp 4716  ‘cfv 5317  1^st c1st 6282  2^nd c2nd 6283  Qcnq 7463  Pcnp 7474  <_P cltp 7478

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 3203  df-ss 3210  df-opab 4145  df-xp 4724  df-iltp 7653

This theorem is referenced by:  ltprordil  7772  ltexprlemm  7783  ltexprlemopl  7784  ltexprlemlol  7785  ltexprlemopu  7786  ltexprlemupu  7787  ltexprlemdisj  7789  ltexprlemloc  7790  ltexprlemfl  7792  ltexprlemrl  7793  ltexprlemfu  7794  ltexprlemru  7795  ltexpri  7796  lteupri  7800  ltaprlem  7801  prplnqu  7803  caucvgprprlemk  7866  caucvgprprlemnkltj  7872  caucvgprprlemnkeqj  7873  caucvgprprlemnjltk  7874  caucvgprprlemnbj  7876  caucvgprprlemml  7877  caucvgprprlemlol  7881  caucvgprprlemupu  7883  suplocexprlemss  7898  suplocexprlemlub  7907  gt0srpr  7931  lttrsr  7945  ltposr  7946  archsr  7965