Theorem ltrelpr 7507

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 7472	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4702	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3189	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2148  ∃wrex 2456   ⊆ wss 3131  {copab 4065   × cxp 4626  ‘cfv 5218  1^st c1st 6142  2^nd c2nd 6143  Qcnq 7282  Pcnp 7293  <_P cltp 7297

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3137  df-ss 3144  df-opab 4067  df-xp 4634  df-iltp 7472

This theorem is referenced by:  ltprordil  7591  ltexprlemm  7602  ltexprlemopl  7603  ltexprlemlol  7604  ltexprlemopu  7605  ltexprlemupu  7606  ltexprlemdisj  7608  ltexprlemloc  7609  ltexprlemfl  7611  ltexprlemrl  7612  ltexprlemfu  7613  ltexprlemru  7614  ltexpri  7615  lteupri  7619  ltaprlem  7620  prplnqu  7622  caucvgprprlemk  7685  caucvgprprlemnkltj  7691  caucvgprprlemnkeqj  7692  caucvgprprlemnjltk  7693  caucvgprprlemnbj  7695  caucvgprprlemml  7696  caucvgprprlemlol  7700  caucvgprprlemupu  7702  suplocexprlemss  7717  suplocexprlemlub  7726  gt0srpr  7750  lttrsr  7764  ltposr  7765  archsr  7784