MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suc11 Structured version   Visualization version   GIF version

Theorem suc11 6287
Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
suc11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Proof of Theorem suc11
StepHypRef Expression
1 eloni 6194 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
2 ordn2lp 6204 . . . . 5 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm3.13 988 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
41, 2, 33syl 18 . . . 4 (𝐴 ∈ On → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
54adantr 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
6 eqimss 4020 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7 sucssel 6276 . . . . . 6 (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵𝐴 ∈ suc 𝐵))
86, 7syl5 34 . . . . 5 (𝐴 ∈ On → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
9 elsuci 6250 . . . . . . 7 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
109ord 858 . . . . . 6 (𝐴 ∈ suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
1110com12 32 . . . . 5 𝐴𝐵 → (𝐴 ∈ suc 𝐵𝐴 = 𝐵))
128, 11syl9 77 . . . 4 (𝐴 ∈ On → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 eqimss2 4021 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14 sucssel 6276 . . . . . 6 (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴𝐵 ∈ suc 𝐴))
1513, 14syl5 34 . . . . 5 (𝐵 ∈ On → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsuci 6250 . . . . . . . 8 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1716ord 858 . . . . . . 7 (𝐵 ∈ suc 𝐴 → (¬ 𝐵𝐴𝐵 = 𝐴))
18 eqcom 2825 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1917, 18syl6ib 252 . . . . . 6 (𝐵 ∈ suc 𝐴 → (¬ 𝐵𝐴𝐴 = 𝐵))
2019com12 32 . . . . 5 𝐵𝐴 → (𝐵 ∈ suc 𝐴𝐴 = 𝐵))
2115, 20syl9 77 . . . 4 (𝐵 ∈ On → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2212, 21jaao 948 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
235, 22mpd 15 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
24 suceq 6249 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2523, 24impbid1 226 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wss 3933  Ord word 6183  Oncon0 6184  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by:  peano4  7593  limenpsi  8680  fin1a2lem2  9811  bnj168  31899  sltval2  33060  sltsolem1  33077  nosepnelem  33081  nolt02o  33096  onsuct0  33686  1oequni2o  34531
  Copyright terms: Public domain W3C validator