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Theorem suc11reg 9079
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 9066 . . . . 5 ¬ (𝐴𝐵𝐵𝐴)
2 ianor 979 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
31, 2mpbi 233 . . . 4 𝐴𝐵 ∨ ¬ 𝐵𝐴)
4 sucidg 6256 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5 eleq2 2904 . . . . . . . . . . 11 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
64, 5syl5ibcom 248 . . . . . . . . . 10 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
7 elsucg 6245 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
86, 7sylibd 242 . . . . . . . . 9 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
98imp 410 . . . . . . . 8 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
109ord 861 . . . . . . 7 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴𝐵𝐴 = 𝐵))
1110ex 416 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵)))
1211com23 86 . . . . 5 (𝐴 ∈ V → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 sucidg 6256 . . . . . . . . . . . 12 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14 eleq2 2904 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
1513, 14syl5ibrcom 250 . . . . . . . . . . 11 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsucg 6245 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
1715, 16sylibd 242 . . . . . . . . . 10 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
1817imp 410 . . . . . . . . 9 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
1918ord 861 . . . . . . . 8 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐵 = 𝐴))
20 eqcom 2831 . . . . . . . 8 (𝐵 = 𝐴𝐴 = 𝐵)
2119, 20syl6ib 254 . . . . . . 7 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐴 = 𝐵))
2221ex 416 . . . . . 6 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵𝐴𝐴 = 𝐵)))
2322com23 86 . . . . 5 (𝐵 ∈ V → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2412, 23jaao 952 . . . 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 7518 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27 sucexb 7518 . . . . . 6 (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
2827notbii 323 . . . . 5 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29 nelneq 2940 . . . . 5 ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3026, 28, 29syl2anb 600 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3130pm2.21d 121 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
32 eqcom 2831 . . . 4 (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
3326notbii 323 . . . . . . 7 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34 nelneq 2940 . . . . . . 7 ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3527, 33, 34syl2anb 600 . . . . . 6 ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3635ancoms 462 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3736pm2.21d 121 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴𝐴 = 𝐵))
3832, 37syl5bi 245 . . 3 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
39 sucprc 6253 . . . . 5 𝐴 ∈ V → suc 𝐴 = 𝐴)
40 sucprc 6253 . . . . 5 𝐵 ∈ V → suc 𝐵 = 𝐵)
4139, 40eqeqan12d 2841 . . . 4 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4241biimpd 232 . . 3 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4325, 31, 38, 424cases 1036 . 2 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
44 suceq 6243 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4543, 44impbii 212 1 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  Vcvv 3480  suc csuc 6180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455  ax-reg 9053
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-eprel 5452  df-fr 5501  df-suc 6184
This theorem is referenced by:  rankxpsuc  9308  unidifsnel  30309  unidifsnne  30310  bnj551  32073  clsk1indlem1  40671
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