Theorem ltrelpr 7648

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 7613	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4762	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3229	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2177  ∃wrex 2486   ⊆ wss 3170  {copab 4115   × cxp 4686  ‘cfv 5285  1^st c1st 6242  2^nd c2nd 6243  Qcnq 7423  Pcnp 7434  <_P cltp 7438

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-in 3176  df-ss 3183  df-opab 4117  df-xp 4694  df-iltp 7613

This theorem is referenced by:  ltprordil  7732  ltexprlemm  7743  ltexprlemopl  7744  ltexprlemlol  7745  ltexprlemopu  7746  ltexprlemupu  7747  ltexprlemdisj  7749  ltexprlemloc  7750  ltexprlemfl  7752  ltexprlemrl  7753  ltexprlemfu  7754  ltexprlemru  7755  ltexpri  7756  lteupri  7760  ltaprlem  7761  prplnqu  7763  caucvgprprlemk  7826  caucvgprprlemnkltj  7832  caucvgprprlemnkeqj  7833  caucvgprprlemnjltk  7834  caucvgprprlemnbj  7836  caucvgprprlemml  7837  caucvgprprlemlol  7841  caucvgprprlemupu  7843  suplocexprlemss  7858  suplocexprlemlub  7867  gt0srpr  7891  lttrsr  7905  ltposr  7906  archsr  7925