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Theorem suc11 6011
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 5918 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
2 ordn2lp 5928 . . . . 5 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm3.13 1017 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
41, 2, 33syl 18 . . . 4 (𝐴 ∈ On → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
54adantr 472 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
6 eqimss 3817 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7 sucssel 6000 . . . . . 6 (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵𝐴 ∈ suc 𝐵))
86, 7syl5 34 . . . . 5 (𝐴 ∈ On → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
9 elsuci 5974 . . . . . . 7 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
109ord 890 . . . . . 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 3818 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14 sucssel 6000 . . . . . 6 (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴𝐵 ∈ suc 𝐴))
1513, 14syl5 34 . . . . 5 (𝐵 ∈ On → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsuci 5974 . . . . . . . 8 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1716ord 890 . . . . . . 7 (𝐵 ∈ suc 𝐴 → (¬ 𝐵𝐴𝐵 = 𝐴))
18 eqcom 2772 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1917, 18syl6ib 242 . . . . . 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 977 . . 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 5973 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2523, 24impbid1 216 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wss 3732  Ord word 5907  Oncon0 5908  suc csuc 5910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-suc 5914
This theorem is referenced by:  peano4  7286  limenpsi  8342  fin1a2lem2  9476  bnj168  31247  sltval2  32253  sltsolem1  32270  nosepnelem  32274  nolt02o  32289  onsuct0  32879  1oequni2o  33649
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