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Mirrors > Home > MPE Home > Th. List > peano4 | Structured version Visualization version GIF version |
Description: Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
peano4 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7877 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | nnon 7877 | . 2 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
3 | suc11 6478 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) | |
4 | 1, 2, 3 | syl2an 594 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Oncon0 6371 suc csuc 6373 ωcom 7871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6374 df-on 6375 df-suc 6377 df-om 7872 |
This theorem is referenced by: dif1ennnALT 9302 fseqdom 10051 finxpreclem4 37001 |
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