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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 8333 | . . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐵 ⊆ 𝐶 ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶))) | |
2 | 1 | 3comr 1127 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ⊆ 𝐶 ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶))) |
3 | nnaword 8333 | . . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ⊆ 𝐵 ↔ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵))) | |
4 | 3 | 3com13 1126 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ⊆ 𝐵 ↔ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵))) |
5 | 2, 4 | anbi12d 634 | . . 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 226 | . 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 3902 | . 2 ⊢ ((𝐴 +o 𝐵) = (𝐴 +o 𝐶) ↔ ((𝐴 +o 𝐵) ⊆ (𝐴 +o 𝐶) ∧ (𝐴 +o 𝐶) ⊆ (𝐴 +o 𝐵))) | |
8 | eqss 3902 | . 2 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) | |
9 | 6, 7, 8 | 3bitr4g 317 | 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 (class class class)co 7191 ωcom 7622 +o coa 8177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-oadd 8184 |
This theorem is referenced by: omopthi 8364 unfilem2 8914 ackbij1lem13 9811 ackbij1lem16 9814 addcanpi 10478 |
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