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Mirrors > Home > MPE Home > Th. List > nnacan | Structured version Visualization version GIF version |
Description: Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnacan | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) = (𝐴 +o 𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnaword 8648 | . . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 ⊆ 𝐶 ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶))) | |
2 | 1 | 3comr 1123 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ⊆ 𝐶 ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶))) |
3 | nnaword 8648 | . . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ⊆ 𝐵 ↔ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵))) | |
4 | 3 | 3com13 1122 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ⊆ 𝐵 ↔ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵))) |
5 | 2, 4 | anbi12d 631 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ ((𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶) ∧ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵)))) |
6 | 5 | bicomd 222 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (((𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶) ∧ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵)) ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))) |
7 | eqss 3995 | . 2 ⊢ ((𝐴 +o 𝐵) = (𝐴 +o 𝐶) ↔ ((𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶) ∧ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵))) | |
8 | eqss 3995 | . 2 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) | |
9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) = (𝐴 +o 𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 (class class class)co 7420 ωcom 7870 +o coa 8484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-oadd 8491 |
This theorem is referenced by: omopthi 8682 nnasmo 8684 unfilem2 9336 ttrcltr 9740 ackbij1lem13 10256 ackbij1lem16 10259 addcanpi 10923 |
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