Theorem ltrelpr 7617

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 7582	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4748	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3224	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2175  ∃wrex 2484   ⊆ wss 3165  {copab 4103   × cxp 4672  ‘cfv 5270  1^st c1st 6223  2^nd c2nd 6224  Qcnq 7392  Pcnp 7403  <_P cltp 7407

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-in 3171  df-ss 3178  df-opab 4105  df-xp 4680  df-iltp 7582

This theorem is referenced by:  ltprordil  7701  ltexprlemm  7712  ltexprlemopl  7713  ltexprlemlol  7714  ltexprlemopu  7715  ltexprlemupu  7716  ltexprlemdisj  7718  ltexprlemloc  7719  ltexprlemfl  7721  ltexprlemrl  7722  ltexprlemfu  7723  ltexprlemru  7724  ltexpri  7725  lteupri  7729  ltaprlem  7730  prplnqu  7732  caucvgprprlemk  7795  caucvgprprlemnkltj  7801  caucvgprprlemnkeqj  7802  caucvgprprlemnjltk  7803  caucvgprprlemnbj  7805  caucvgprprlemml  7806  caucvgprprlemlol  7810  caucvgprprlemupu  7812  suplocexprlemss  7827  suplocexprlemlub  7836  gt0srpr  7860  lttrsr  7874  ltposr  7875  archsr  7894