Theorem lerel 10102

Description: 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
lerel	⊢ Rel ≤

Proof of Theorem lerel

Step	Hyp	Ref	Expression
1		lerelxr 10101	. 2 ⊢ ≤ ⊆ (ℝ* × ℝ*)
2		relxp 5227	. 2 ⊢ Rel (ℝ* × ℝ*)
3		relss 5206	. 2 ⊢ ( ≤ ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel ≤ ))
4	1, 2, 3	mp2 9	1 ⊢ Rel ≤

Syntax hints:   ⊆ wss 3574   × cxp 5112  Rel wrel 5119  ℝ^*cxr 10073   ≤ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120  df-rel 5121  df-le 10080

This theorem is referenced by:  dfle2  11980  dflt2  11981  ledm  17224  lern  17225  lefld  17226  letsr  17227  dvle  23770  gtiso  29478