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, B⟫ ∈ ≤fin ↔ (⟪A, B⟫ ∈ <fin ∨ A = B)))

Proof of Theorem lefinlteq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	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
4	2, 3	syl6req 2402	. . . . . . . . 9 ⊢ (x = 0c → A = (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))
7	5, 6	syl6eqr 2403	. . . . . . . . . 10 ⊢ (x = (y +c 1c) → (A +c x) = ((A +c y) +c 1c))
8	7	reximi 2721	. . . . . . . . 9 ⊢ (∃y ∈ Nn x = (y +c 1c) → ∃y ∈ Nn (A +c x) = ((A +c y) +c 1c))
9	4, 8	orim12i 502	. . . . . . . 8 ⊢ ((x = 0c ∨ ∃y ∈ Nn x = (y +c 1c)) → (A = (A +c x) ∨ ∃y ∈ Nn (A +c x) = ((A +c y) +c 1c)))
10	1, 9	sylbi 187	. . . . . . 7 ⊢ (x ∈ Nn → (A = (A +c x) ∨ ∃y ∈ Nn (A +c x) = ((A +c y) +c 1c)))
11	10	orcomd 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)))
13	12	rexbidv 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 = B ↔ A = (A +c x)))
15	13, 14	orbi12d 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))))
16	11, 15	syl5ibrcom 213	. . . . 5 ⊢ (x ∈ Nn → (B = (A +c x) → (∃y ∈ Nn B = ((A +c y) +c 1c) ∨ A = B)))
17	16	rexlimiv 2732	. . . 4 ⊢ (∃x ∈ Nn B = (A +c x) → (∃y ∈ Nn B = ((A +c y) +c 1c) ∨ A = B))
18	6	eqeq2i 2363	. . . . . . 7 ⊢ (B = ((A +c y) +c 1c) ↔ B = (A +c (y +c 1c)))
19		peano2 4403	. . . . . . . 8 ⊢ (y ∈ Nn → (y +c 1c) ∈ Nn )
20	5	eqeq2d 2364	. . . . . . . . 9 ⊢ (x = (y +c 1c) → (B = (A +c x) ↔ B = (A +c (y +c 1c))))
21	20	rspcev 2955	. . . . . . . 8 ⊢ (((y +c 1c) ∈ Nn ∧ B = (A +c (y +c 1c))) → ∃x ∈ Nn B = (A +c x))
22	19, 21	sylan 457	. . . . . . 7 ⊢ ((y ∈ Nn ∧ B = (A +c (y +c 1c))) → ∃x ∈ Nn B = (A +c x))
23	18, 22	sylan2b 461	. . . . . 6 ⊢ ((y ∈ Nn ∧ B = ((A +c y) +c 1c)) → ∃x ∈ Nn B = (A +c x))
24	23	rexlimiva 2733	. . . . 5 ⊢ (∃y ∈ Nn B = ((A +c y) +c 1c) → ∃x ∈ Nn B = (A +c x))
25		peano1 4402	. . . . . . 7 ⊢ 0c ∈ Nn
26	3	eqcomi 2357	. . . . . . 7 ⊢ A = (A +c 0c)
27	2	eqeq2d 2364	. . . . . . . 8 ⊢ (x = 0c → (A = (A +c x) ↔ A = (A +c 0c)))
28	27	rspcev 2955	. . . . . . 7 ⊢ ((0c ∈ Nn ∧ A = (A +c 0c)) → ∃x ∈ Nn A = (A +c x))
29	25, 26, 28	mp2an 653	. . . . . 6 ⊢ ∃x ∈ Nn A = (A +c x)
30		eqeq1 2359	. . . . . . 7 ⊢ (A = B → (A = (A +c x) ↔ B = (A +c x)))
31	30	rexbidv 2635	. . . . . 6 ⊢ (A = B → (∃x ∈ Nn A = (A +c x) ↔ ∃x ∈ Nn B = (A +c x)))
32	29, 31	mpbii 202	. . . . 5 ⊢ (A = B → ∃x ∈ Nn B = (A +c x))
33	24, 32	jaoi 368	. . . 4 ⊢ ((∃y ∈ Nn B = ((A +c y) +c 1c) ∨ A = B) → ∃x ∈ Nn B = (A +c x))
34	17, 33	impbii 180	. . 3 ⊢ (∃x ∈ Nn B = (A +c x) ↔ (∃y ∈ Nn B = ((A +c y) +c 1c) ∨ A = B))
35	34	a1i 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, B⟫ ∈ ≤fin ↔ ∃x ∈ Nn B = (A +c x)))
37	36	3adant3 975	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ A ≠ ∅) → (⟪A, B⟫ ∈ ≤fin ↔ ∃x ∈ 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))))
39	38	adantr 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))))
41	40	adantl 452	. . . . 5 ⊢ (((A ∈ V ∧ B ∈ W) ∧ A ≠ ∅) → (∃y ∈ Nn B = ((A +c y) +c 1c) ↔ (A ≠ ∅ ∧ ∃y ∈ Nn B = ((A +c y) +c 1c))))
42	39, 41	bitr4d 247	. . . 4 ⊢ (((A ∈ V ∧ B ∈ W) ∧ A ≠ ∅) → (⟪A, B⟫ ∈ <fin ↔ ∃y ∈ Nn B = ((A +c y) +c 1c)))
43	42	orbi1d 683	. . 3 ⊢ (((A ∈ V ∧ B ∈ W) ∧ A ≠ ∅) → ((⟪A, B⟫ ∈ <fin ∨ A = B) ↔ (∃y ∈ Nn B = ((A +c y) +c 1c) ∨ A = B)))
44	43	3impa 1146	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ A ≠ ∅) → ((⟪A, B⟫ ∈ <fin ∨ A = B) ↔ (∃y ∈ Nn B = ((A +c y) +c 1c) ∨ A = B)))
45	35, 37, 44	3bitr4d 276	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ A ≠ ∅) → (⟪A, B⟫ ∈ ≤fin ↔ (⟪A, B⟫ ∈ <fin ∨ A = B)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ⟪copk 4057  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +_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-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-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-0c 4377  df-addc 4378  df-nnc 4379  df-lefin 4440  df-ltfin 4441

This theorem is referenced by:  ltfintri  4466  vfin1cltv  4547