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, B⟫ ∈ ≤fin ))

Proof of Theorem ltlefin

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

Step	Hyp	Ref	Expression
1		addcass 4415	. . . . . . 7 ⊢ ((A +c x) +c 1c) = (A +c (x +c 1c))
2	1	eqeq2i 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)))
5	4	eqeq2d 2364	. . . . . . . 8 ⊢ (y = (x +c 1c) → (B = (A +c y) ↔ B = (A +c (x +c 1c))))
6	5	rspcev 2955	. . . . . . 7 ⊢ (((x +c 1c) ∈ Nn ∧ B = (A +c (x +c 1c))) → ∃y ∈ Nn B = (A +c y))
7	3, 6	sylan 457	. . . . . 6 ⊢ ((x ∈ Nn ∧ B = (A +c (x +c 1c))) → ∃y ∈ Nn B = (A +c y))
8	2, 7	sylan2b 461	. . . . 5 ⊢ ((x ∈ Nn ∧ B = ((A +c x) +c 1c)) → ∃y ∈ Nn B = (A +c y))
9	8	rexlimiva 2733	. . . 4 ⊢ (∃x ∈ Nn B = ((A +c x) +c 1c) → ∃y ∈ Nn B = (A +c y))
10	9	adantl 452	. . 3 ⊢ ((A ≠ ∅ ∧ ∃x ∈ Nn B = ((A +c x) +c 1c)) → ∃y ∈ Nn B = (A +c y))
11	10	a1i 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, B⟫ ∈ ≤fin ↔ ∃y ∈ Nn B = (A +c y)))
14	11, 12, 13	3imtr4d 259	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin ))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ⟪copk 4057  1_cc1c 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