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Theorem suc11 5734
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 5636 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
2 ordn2lp 5646 . . . . 5 (Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
3 pm3.13 521 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
41, 2, 33syl 18 . . . 4 (𝐴 ∈ On → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
54adantr 480 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
6 eqimss 3620 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7 sucssel 5722 . . . . . 6 (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵𝐴 ∈ suc 𝐵))
86, 7syl5 33 . . . . 5 (𝐴 ∈ On → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
9 elsuci 5694 . . . . . . 7 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
109ord 391 . . . . . 6 (𝐴 ∈ suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
1110com12 32 . . . . 5 𝐴𝐵 → (𝐴 ∈ suc 𝐵𝐴 = 𝐵))
128, 11syl9 75 . . . 4 (𝐴 ∈ On → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 eqimss2 3621 . . . . . 6 (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14 sucssel 5722 . . . . . 6 (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴𝐵 ∈ suc 𝐴))
1513, 14syl5 33 . . . . 5 (𝐵 ∈ On → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsuci 5694 . . . . . . . 8 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
1716ord 391 . . . . . . 7 (𝐵 ∈ suc 𝐴 → (¬ 𝐵𝐴𝐵 = 𝐴))
18 eqcom 2617 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1917, 18syl6ib 240 . . . . . 6 (𝐵 ∈ suc 𝐴 → (¬ 𝐵𝐴𝐴 = 𝐵))
2019com12 32 . . . . 5 𝐵𝐴 → (𝐵 ∈ suc 𝐴𝐴 = 𝐵))
2115, 20syl9 75 . . . 4 (𝐵 ∈ On → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2212, 21jaao 530 . . 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 5693 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2523, 24impbid1 214 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wss 3540  Ord word 5625  Oncon0 5626  suc csuc 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-tr 4676  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-suc 5632
This theorem is referenced by:  peano4  6958  limenpsi  7998  fin1a2lem2  9084  bnj168  29846  sltval2  30847  sltsolem1  30861  onsuct0  31404
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