Theorem lenltfin 4470

Description: Less than or equal is the same as negated less than. (Contributed by SF, 2-Feb-2015.)

Assertion

Ref	Expression
lenltfin	⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (⟪A, B⟫ ∈ ≤fin ↔ ¬ ⟪B, A⟫ ∈ <fin ))

Proof of Theorem lenltfin

Step	Hyp	Ref	Expression
1		ltfinirr 4458	. . . . . 6 ⊢ (A ∈ Nn → ¬ ⟪A, A⟫ ∈ <fin )
2	1	adantr 451	. . . . 5 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → ¬ ⟪A, A⟫ ∈ <fin )
3	2	adantr 451	. . . 4 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ ⟪A, B⟫ ∈ ≤fin ) → ¬ ⟪A, A⟫ ∈ <fin )
4		leltfintr 4459	. . . . . 6 ⊢ ((A ∈ Nn ∧ B ∈ Nn ∧ A ∈ Nn ) → ((⟪A, B⟫ ∈ ≤fin ∧ ⟪B, A⟫ ∈ <fin ) → ⟪A, A⟫ ∈ <fin ))
5	4	3anidm13 1240	. . . . 5 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → ((⟪A, B⟫ ∈ ≤fin ∧ ⟪B, A⟫ ∈ <fin ) → ⟪A, A⟫ ∈ <fin ))
6	5	expdimp 426	. . . 4 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ ⟪A, B⟫ ∈ ≤fin ) → (⟪B, A⟫ ∈ <fin → ⟪A, A⟫ ∈ <fin ))
7	3, 6	mtod 168	. . 3 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ ⟪A, B⟫ ∈ ≤fin ) → ¬ ⟪B, A⟫ ∈ <fin )
8	7	ex 423	. 2 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (⟪A, B⟫ ∈ ≤fin → ¬ ⟪B, A⟫ ∈ <fin ))
9		nulge 4457	. . . . . 6 ⊢ ((∅ ∈ Nn ∧ A ∈ Nn ) → ⟪A, ∅⟫ ∈ ≤fin )
10	9	ancoms 439	. . . . 5 ⊢ ((A ∈ Nn ∧ ∅ ∈ Nn ) → ⟪A, ∅⟫ ∈ ≤fin )
11		eleq1 2413	. . . . . . 7 ⊢ (B = ∅ → (B ∈ Nn ↔ ∅ ∈ Nn ))
12	11	anbi2d 684	. . . . . 6 ⊢ (B = ∅ → ((A ∈ Nn ∧ B ∈ Nn ) ↔ (A ∈ Nn ∧ ∅ ∈ Nn )))
13		opkeq2 4061	. . . . . . 7 ⊢ (B = ∅ → ⟪A, B⟫ = ⟪A, ∅⟫)
14	13	eleq1d 2419	. . . . . 6 ⊢ (B = ∅ → (⟪A, B⟫ ∈ ≤fin ↔ ⟪A, ∅⟫ ∈ ≤fin ))
15	12, 14	imbi12d 311	. . . . 5 ⊢ (B = ∅ → (((A ∈ Nn ∧ B ∈ Nn ) → ⟪A, B⟫ ∈ ≤fin ) ↔ ((A ∈ Nn ∧ ∅ ∈ Nn ) → ⟪A, ∅⟫ ∈ ≤fin )))
16	10, 15	mpbiri 224	. . . 4 ⊢ (B = ∅ → ((A ∈ Nn ∧ B ∈ Nn ) → ⟪A, B⟫ ∈ ≤fin ))
17	16	a1dd 42	. . 3 ⊢ (B = ∅ → ((A ∈ Nn ∧ B ∈ Nn ) → (¬ ⟪B, A⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin )))
18		simplr 731	. . . . . . . 8 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → B ∈ Nn )
19		simpll 730	. . . . . . . 8 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → A ∈ Nn )
20		simpr 447	. . . . . . . 8 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → B ≠ ∅)
21		ltfintri 4467	. . . . . . . 8 ⊢ ((B ∈ Nn ∧ A ∈ Nn ∧ B ≠ ∅) → (⟪B, A⟫ ∈ <fin ∨ B = A ∨ ⟪A, B⟫ ∈ <fin ))
22	18, 19, 20, 21	syl3anc 1182	. . . . . . 7 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → (⟪B, A⟫ ∈ <fin ∨ B = A ∨ ⟪A, B⟫ ∈ <fin ))
23		3orass 937	. . . . . . 7 ⊢ ((⟪B, A⟫ ∈ <fin ∨ B = A ∨ ⟪A, B⟫ ∈ <fin ) ↔ (⟪B, A⟫ ∈ <fin ∨ (B = A ∨ ⟪A, B⟫ ∈ <fin )))
24	22, 23	sylib 188	. . . . . 6 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → (⟪B, A⟫ ∈ <fin ∨ (B = A ∨ ⟪A, B⟫ ∈ <fin )))
25	24	ord 366	. . . . 5 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → (¬ ⟪B, A⟫ ∈ <fin → (B = A ∨ ⟪A, B⟫ ∈ <fin )))
26		lefinrflx 4468	. . . . . . . . 9 ⊢ (A ∈ Nn → ⟪A, A⟫ ∈ ≤fin )
27	26	adantr 451	. . . . . . . 8 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → ⟪A, A⟫ ∈ ≤fin )
28		opkeq2 4061	. . . . . . . . 9 ⊢ (B = A → ⟪A, B⟫ = ⟪A, A⟫)
29	28	eleq1d 2419	. . . . . . . 8 ⊢ (B = A → (⟪A, B⟫ ∈ ≤fin ↔ ⟪A, A⟫ ∈ ≤fin ))
30	27, 29	syl5ibrcom 213	. . . . . . 7 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (B = A → ⟪A, B⟫ ∈ ≤fin ))
31	30	adantr 451	. . . . . 6 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → (B = A → ⟪A, B⟫ ∈ ≤fin ))
32		ltlefin 4469	. . . . . . 7 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (⟪A, B⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin ))
33	32	adantr 451	. . . . . 6 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → (⟪A, B⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin ))
34	31, 33	jaod 369	. . . . 5 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → ((B = A ∨ ⟪A, B⟫ ∈ <fin ) → ⟪A, B⟫ ∈ ≤fin ))
35	25, 34	syld 40	. . . 4 ⊢ (((A ∈ Nn ∧ B ∈ Nn ) ∧ B ≠ ∅) → (¬ ⟪B, A⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin ))
36	35	expcom 424	. . 3 ⊢ (B ≠ ∅ → ((A ∈ Nn ∧ B ∈ Nn ) → (¬ ⟪B, A⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin )))
37	17, 36	pm2.61ine 2593	. 2 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (¬ ⟪B, A⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin ))
38	8, 37	impbid 183	1 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (⟪A, B⟫ ∈ ≤fin ↔ ¬ ⟪B, A⟫ ∈ <fin ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∅c0 3551  ⟪copk 4058   Nn cnnc 4374   ≤_fin clefin 4433   <_fin cltfin 4434

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380  df-lefin 4441  df-ltfin 4442

This theorem is used by:  tfinlefin  4503  vfinspsslem1  4551