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Theorem suc11reg 8735
Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
Assertion
Ref Expression
suc11reg (suc 𝐴 = suc 𝐵𝐴 = 𝐵)

Proof of Theorem suc11reg
StepHypRef Expression
1 en2lp 8721 . . . . 5 ¬ (𝐴𝐵𝐵𝐴)
2 ianor 1004 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
31, 2mpbi 221 . . . 4 𝐴𝐵 ∨ ¬ 𝐵𝐴)
4 sucidg 5988 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5 eleq2 2833 . . . . . . . . . . 11 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
64, 5syl5ibcom 236 . . . . . . . . . 10 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
7 elsucg 5977 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
86, 7sylibd 230 . . . . . . . . 9 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
98imp 395 . . . . . . . 8 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
109ord 890 . . . . . . 7 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴𝐵𝐴 = 𝐵))
1110ex 401 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵)))
1211com23 86 . . . . 5 (𝐴 ∈ V → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 sucidg 5988 . . . . . . . . . . . 12 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14 eleq2 2833 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
1513, 14syl5ibrcom 238 . . . . . . . . . . 11 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsucg 5977 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
1715, 16sylibd 230 . . . . . . . . . 10 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
1817imp 395 . . . . . . . . 9 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
1918ord 890 . . . . . . . 8 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐵 = 𝐴))
20 eqcom 2772 . . . . . . . 8 (𝐵 = 𝐴𝐴 = 𝐵)
2119, 20syl6ib 242 . . . . . . 7 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐴 = 𝐵))
2221ex 401 . . . . . 6 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵𝐴𝐴 = 𝐵)))
2322com23 86 . . . . 5 (𝐵 ∈ V → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2412, 23jaao 977 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
253, 24mpi 20 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
26 sucexb 7211 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27 sucexb 7211 . . . . . 6 (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
2827notbii 311 . . . . 5 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29 nelneq 2868 . . . . 5 ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3026, 28, 29syl2anb 591 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3130pm2.21d 119 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
32 eqcom 2772 . . . 4 (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
3326notbii 311 . . . . . . 7 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34 nelneq 2868 . . . . . . 7 ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3527, 33, 34syl2anb 591 . . . . . 6 ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3635ancoms 450 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3736pm2.21d 119 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴𝐴 = 𝐵))
3832, 37syl5bi 233 . . 3 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
39 sucprc 5985 . . . . 5 𝐴 ∈ V → suc 𝐴 = 𝐴)
40 sucprc 5985 . . . . 5 𝐵 ∈ V → suc 𝐵 = 𝐵)
4139, 40eqeqan12d 2781 . . . 4 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4241biimpd 220 . . 3 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4325, 31, 38, 424cases 1063 . 2 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
44 suceq 5975 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4543, 44impbii 200 1 (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  Vcvv 3350  suc csuc 5912
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-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151  ax-reg 8708
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 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-eprel 5192  df-fr 5238  df-suc 5916
This theorem is referenced by:  rankxpsuc  8964  bnj551  31281  clsk1indlem1  39041
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