Theorem ltrelpr 7214

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 7179	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4551	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3079	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 103   ∈ wcel 1448  ∃wrex 2376   ⊆ wss 3021  {copab 3928   × cxp 4475  ‘cfv 5059  1^st c1st 5967  2^nd c2nd 5968  Qcnq 6989  Pcnp 7000  <_P cltp 7004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-in 3027  df-ss 3034  df-opab 3930  df-xp 4483  df-iltp 7179

This theorem is referenced by:  ltprordil  7298  ltexprlemm  7309  ltexprlemopl  7310  ltexprlemlol  7311  ltexprlemopu  7312  ltexprlemupu  7313  ltexprlemdisj  7315  ltexprlemloc  7316  ltexprlemfl  7318  ltexprlemrl  7319  ltexprlemfu  7320  ltexprlemru  7321  ltexpri  7322  lteupri  7326  ltaprlem  7327  prplnqu  7329  caucvgprprlemk  7392  caucvgprprlemnkltj  7398  caucvgprprlemnkeqj  7399  caucvgprprlemnjltk  7400  caucvgprprlemnbj  7402  caucvgprprlemml  7403  caucvgprprlemlol  7407  caucvgprprlemupu  7409  gt0srpr  7444  lttrsr  7458  ltposr  7459  archsr  7477