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Theorem lefinrflx 4467
 Description: Less than or equal to is reflexive. (Contributed by SF, 2-Feb-2015.)
Assertion
Ref Expression
lefinrflx (A V → ⟪A, Afin )

Proof of Theorem lefinrflx
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 peano1 4402 . . 3 0c Nn
2 addcid1 4405 . . . 4 (A +c 0c) = A
32eqcomi 2357 . . 3 A = (A +c 0c)
4 addceq2 4384 . . . . 5 (x = 0c → (A +c x) = (A +c 0c))
54eqeq2d 2364 . . . 4 (x = 0c → (A = (A +c x) ↔ A = (A +c 0c)))
65rspcev 2955 . . 3 ((0c Nn A = (A +c 0c)) → x Nn A = (A +c x))
71, 3, 6mp2an 653 . 2 x Nn A = (A +c x)
8 opklefing 4448 . . 3 ((A V A V) → (⟪A, Afinx Nn A = (A +c x)))
98anidms 626 . 2 (A V → (⟪A, Afinx Nn A = (A +c x)))
107, 9mpbiri 224 1 (A V → ⟪A, Afin )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟪copk 4057   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   ≤fin clefin 4432 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378  df-nnc 4379  df-lefin 4440 This theorem is referenced by:  lenltfin  4469
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