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Theorem suc11 6472
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 6375 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
2 ordn2lp 6385 . . . . 5 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm3.13 994 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
41, 2, 33syl 18 . . . 4 (𝐴 ∈ On → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
54adantr 482 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
6 eqimss 4041 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7 sucssel 6460 . . . . . 6 (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵𝐴 ∈ suc 𝐵))
86, 7syl5 34 . . . . 5 (𝐴 ∈ On → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
9 elsuci 6432 . . . . . . 7 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
109ord 863 . . . . . 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 4042 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14 sucssel 6460 . . . . . 6 (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴𝐵 ∈ suc 𝐴))
1513, 14syl5 34 . . . . 5 (𝐵 ∈ On → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsuci 6432 . . . . . . . 8 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1716ord 863 . . . . . . 7 (𝐵 ∈ suc 𝐴 → (¬ 𝐵𝐴𝐵 = 𝐴))
18 eqcom 2740 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1917, 18imbitrdi 250 . . . . . 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 954 . . 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 6431 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2523, 24impbid1 224 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wss 3949  Ord word 6364  Oncon0 6365  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-suc 6371
This theorem is referenced by:  peano4  7883  limenpsi  9152  fin1a2lem2  10396  sltval2  27159  sltsolem1  27178  nosepnelem  27182  nolt02o  27198  bnj168  33741  onsuct0  35326  1oequni2o  36249  onsucf1lem  42019  onsucf1o  42022
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