Theorem ltrelpr 7503

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 7468	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4700	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 3187	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 104   ∈ wcel 2148  ∃wrex 2456   ⊆ wss 3129  {copab 4063   × cxp 4624  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139  Qcnq 7278  Pcnp 7289  <_P cltp 7293

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 3135  df-ss 3142  df-opab 4065  df-xp 4632  df-iltp 7468

This theorem is referenced by:  ltprordil  7587  ltexprlemm  7598  ltexprlemopl  7599  ltexprlemlol  7600  ltexprlemopu  7601  ltexprlemupu  7602  ltexprlemdisj  7604  ltexprlemloc  7605  ltexprlemfl  7607  ltexprlemrl  7608  ltexprlemfu  7609  ltexprlemru  7610  ltexpri  7611  lteupri  7615  ltaprlem  7616  prplnqu  7618  caucvgprprlemk  7681  caucvgprprlemnkltj  7687  caucvgprprlemnkeqj  7688  caucvgprprlemnjltk  7689  caucvgprprlemnbj  7691  caucvgprprlemml  7692  caucvgprprlemlol  7696  caucvgprprlemupu  7698  suplocexprlemss  7713  suplocexprlemlub  7722  gt0srpr  7746  lttrsr  7760  ltposr  7761  archsr  7780