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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 7879 | . . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 ⊆ 𝐶 ↔ (𝐴 +𝑜 𝐵) ⊆ (𝐴 +𝑜 𝐶))) | |
2 | 1 | 3comr 1120 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ⊆ 𝐶 ↔ (𝐴 +𝑜 𝐵) ⊆ (𝐴 +𝑜 𝐶))) |
3 | nnaword 7879 | . . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ⊆ 𝐵 ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐴 +𝑜 𝐵))) | |
4 | 3 | 3com13 1119 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ⊆ 𝐵 ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐴 +𝑜 𝐵))) |
5 | 2, 4 | anbi12d 749 | . . 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 213 | . 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 3760 | . 2 ⊢ ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ ((𝐴 +𝑜 𝐵) ⊆ (𝐴 +𝑜 𝐶) ∧ (𝐴 +𝑜 𝐶) ⊆ (𝐴 +𝑜 𝐵))) | |
8 | eqss 3760 | . 2 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) | |
9 | 6, 7, 8 | 3bitr4g 303 | 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 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2140 ⊆ wss 3716 (class class class)co 6815 ωcom 7232 +𝑜 coa 7728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-oadd 7735 |
This theorem is referenced by: omopthi 7909 unfilem2 8393 ackbij1lem13 9267 ackbij1lem16 9270 addcanpi 9934 |
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