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Theorem lefinlteq 4463
 Description: Transfer from less than or equal to less than. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
lefinlteq ((A V B W A) → (⟪A, Bfin ↔ (⟪A, B <fin A = B)))

Proof of Theorem lefinlteq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnc0suc 4412 . . . . . . . 8 (x Nn ↔ (x = 0c y Nn x = (y +c 1c)))
2 addceq2 4384 . . . . . . . . . 10 (x = 0c → (A +c x) = (A +c 0c))
3 addcid1 4405 . . . . . . . . . 10 (A +c 0c) = A
42, 3syl6req 2402 . . . . . . . . 9 (x = 0cA = (A +c x))
5 addceq2 4384 . . . . . . . . . . 11 (x = (y +c 1c) → (A +c x) = (A +c (y +c 1c)))
6 addcass 4415 . . . . . . . . . . 11 ((A +c y) +c 1c) = (A +c (y +c 1c))
75, 6syl6eqr 2403 . . . . . . . . . 10 (x = (y +c 1c) → (A +c x) = ((A +c y) +c 1c))
87reximi 2721 . . . . . . . . 9 (y Nn x = (y +c 1c) → y Nn (A +c x) = ((A +c y) +c 1c))
94, 8orim12i 502 . . . . . . . 8 ((x = 0c y Nn x = (y +c 1c)) → (A = (A +c x) y Nn (A +c x) = ((A +c y) +c 1c)))
101, 9sylbi 187 . . . . . . 7 (x Nn → (A = (A +c x) y Nn (A +c x) = ((A +c y) +c 1c)))
1110orcomd 377 . . . . . 6 (x Nn → (y Nn (A +c x) = ((A +c y) +c 1c) A = (A +c x)))
12 eqeq1 2359 . . . . . . . 8 (B = (A +c x) → (B = ((A +c y) +c 1c) ↔ (A +c x) = ((A +c y) +c 1c)))
1312rexbidv 2635 . . . . . . 7 (B = (A +c x) → (y Nn B = ((A +c y) +c 1c) ↔ y Nn (A +c x) = ((A +c y) +c 1c)))
14 eqeq2 2362 . . . . . . 7 (B = (A +c x) → (A = BA = (A +c x)))
1513, 14orbi12d 690 . . . . . 6 (B = (A +c x) → ((y Nn B = ((A +c y) +c 1c) A = B) ↔ (y Nn (A +c x) = ((A +c y) +c 1c) A = (A +c x))))
1611, 15syl5ibrcom 213 . . . . 5 (x Nn → (B = (A +c x) → (y Nn B = ((A +c y) +c 1c) A = B)))
1716rexlimiv 2732 . . . 4 (x Nn B = (A +c x) → (y Nn B = ((A +c y) +c 1c) A = B))
186eqeq2i 2363 . . . . . . 7 (B = ((A +c y) +c 1c) ↔ B = (A +c (y +c 1c)))
19 peano2 4403 . . . . . . . 8 (y Nn → (y +c 1c) Nn )
205eqeq2d 2364 . . . . . . . . 9 (x = (y +c 1c) → (B = (A +c x) ↔ B = (A +c (y +c 1c))))
2120rspcev 2955 . . . . . . . 8 (((y +c 1c) Nn B = (A +c (y +c 1c))) → x Nn B = (A +c x))
2219, 21sylan 457 . . . . . . 7 ((y Nn B = (A +c (y +c 1c))) → x Nn B = (A +c x))
2318, 22sylan2b 461 . . . . . 6 ((y Nn B = ((A +c y) +c 1c)) → x Nn B = (A +c x))
2423rexlimiva 2733 . . . . 5 (y Nn B = ((A +c y) +c 1c) → x Nn B = (A +c x))
25 peano1 4402 . . . . . . 7 0c Nn
263eqcomi 2357 . . . . . . 7 A = (A +c 0c)
272eqeq2d 2364 . . . . . . . 8 (x = 0c → (A = (A +c x) ↔ A = (A +c 0c)))
2827rspcev 2955 . . . . . . 7 ((0c Nn A = (A +c 0c)) → x Nn A = (A +c x))
2925, 26, 28mp2an 653 . . . . . 6 x Nn A = (A +c x)
30 eqeq1 2359 . . . . . . 7 (A = B → (A = (A +c x) ↔ B = (A +c x)))
3130rexbidv 2635 . . . . . 6 (A = B → (x Nn A = (A +c x) ↔ x Nn B = (A +c x)))
3229, 31mpbii 202 . . . . 5 (A = Bx Nn B = (A +c x))
3324, 32jaoi 368 . . . 4 ((y Nn B = ((A +c y) +c 1c) A = B) → x Nn B = (A +c x))
3417, 33impbii 180 . . 3 (x Nn B = (A +c x) ↔ (y Nn B = ((A +c y) +c 1c) A = B))
3534a1i 10 . 2 ((A V B W A) → (x Nn B = (A +c x) ↔ (y Nn B = ((A +c y) +c 1c) A = B)))
36 opklefing 4448 . . 3 ((A V B W) → (⟪A, Bfinx Nn B = (A +c x)))
37363adant3 975 . 2 ((A V B W A) → (⟪A, Bfinx Nn B = (A +c x)))
38 opkltfing 4449 . . . . . 6 ((A V B W) → (⟪A, B <fin ↔ (A y Nn B = ((A +c y) +c 1c))))
3938adantr 451 . . . . 5 (((A V B W) A) → (⟪A, B <fin ↔ (A y Nn B = ((A +c y) +c 1c))))
40 ibar 490 . . . . . 6 (A → (y Nn B = ((A +c y) +c 1c) ↔ (A y Nn B = ((A +c y) +c 1c))))
4140adantl 452 . . . . 5 (((A V B W) A) → (y Nn B = ((A +c y) +c 1c) ↔ (A y Nn B = ((A +c y) +c 1c))))
4239, 41bitr4d 247 . . . 4 (((A V B W) A) → (⟪A, B <finy Nn B = ((A +c y) +c 1c)))
4342orbi1d 683 . . 3 (((A V B W) A) → ((⟪A, B <fin A = B) ↔ (y Nn B = ((A +c y) +c 1c) A = B)))
44433impa 1146 . 2 ((A V B W A) → ((⟪A, B <fin A = B) ↔ (y Nn B = ((A +c y) +c 1c) A = B)))
4535, 37, 443bitr4d 276 1 ((A V B W A) → (⟪A, Bfin ↔ (⟪A, B <fin A = B)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ⟪copk 4057  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   ≤fin clefin 4432
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