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Theorem ltlefin 4468
Description: Less than implies less than or equal. (Contributed by SF, 2-Feb-2015.)
Assertion
Ref Expression
ltlefin ((A V B W) → (⟪A, B <fin → ⟪A, Bfin ))

Proof of Theorem ltlefin
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcass 4415 . . . . . . 7 ((A +c x) +c 1c) = (A +c (x +c 1c))
21eqeq2i 2363 . . . . . 6 (B = ((A +c x) +c 1c) ↔ B = (A +c (x +c 1c)))
3 peano2 4403 . . . . . . 7 (x Nn → (x +c 1c) Nn )
4 addceq2 4384 . . . . . . . . 9 (y = (x +c 1c) → (A +c y) = (A +c (x +c 1c)))
54eqeq2d 2364 . . . . . . . 8 (y = (x +c 1c) → (B = (A +c y) ↔ B = (A +c (x +c 1c))))
65rspcev 2955 . . . . . . 7 (((x +c 1c) Nn B = (A +c (x +c 1c))) → y Nn B = (A +c y))
73, 6sylan 457 . . . . . 6 ((x Nn B = (A +c (x +c 1c))) → y Nn B = (A +c y))
82, 7sylan2b 461 . . . . 5 ((x Nn B = ((A +c x) +c 1c)) → y Nn B = (A +c y))
98rexlimiva 2733 . . . 4 (x Nn B = ((A +c x) +c 1c) → y Nn B = (A +c y))
109adantl 452 . . 3 ((A x Nn B = ((A +c x) +c 1c)) → y Nn B = (A +c y))
1110a1i 10 . 2 ((A V B W) → ((A x Nn B = ((A +c x) +c 1c)) → y Nn B = (A +c y)))
12 opkltfing 4449 . 2 ((A V B W) → (⟪A, B <fin ↔ (A x Nn B = ((A +c x) +c 1c))))
13 opklefing 4448 . 2 ((A V B W) → (⟪A, Bfiny Nn B = (A +c y)))
1411, 12, 133imtr4d 259 1 ((A V B W) → (⟪A, B <fin → ⟪A, Bfin ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358   = wceq 1642   wcel 1710  wne 2516  wrex 2615  c0 3550  copk 4057  1cc1c 4134   Nn cnnc 4373   +c cplc 4375  fin clefin 4432   <fin cltfin 4433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  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-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-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-addc 4378  df-nnc 4379  df-lefin 4440  df-ltfin 4441
This theorem is referenced by:  lenltfin  4469
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