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Mirrors > Home > NFE Home > Th. List > ltlefin | GIF version |
Description: Less than implies less than or equal. (Contributed by SF, 2-Feb-2015.) |
Ref | Expression |
---|---|
ltlefin | ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ <fin → ⟪A, B⟫ ∈ ≤fin )) |
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 )) |
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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