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Theorem suc11reg 9563
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 9550 . . . . 5 ¬ (𝐴𝐵𝐵𝐴)
2 ianor 981 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
31, 2mpbi 229 . . . 4 𝐴𝐵 ∨ ¬ 𝐵𝐴)
4 sucidg 6402 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5 eleq2 2823 . . . . . . . . . . 11 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
64, 5syl5ibcom 244 . . . . . . . . . 10 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
7 elsucg 6389 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
86, 7sylibd 238 . . . . . . . . 9 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
98imp 408 . . . . . . . 8 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
109ord 863 . . . . . . 7 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴𝐵𝐴 = 𝐵))
1110ex 414 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵)))
1211com23 86 . . . . 5 (𝐴 ∈ V → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 sucidg 6402 . . . . . . . . . . . 12 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14 eleq2 2823 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
1513, 14syl5ibrcom 247 . . . . . . . . . . 11 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsucg 6389 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
1715, 16sylibd 238 . . . . . . . . . 10 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
1817imp 408 . . . . . . . . 9 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
1918ord 863 . . . . . . . 8 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐵 = 𝐴))
20 eqcom 2740 . . . . . . . 8 (𝐵 = 𝐴𝐴 = 𝐵)
2119, 20syl6ib 251 . . . . . . 7 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐴 = 𝐵))
2221ex 414 . . . . . 6 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵𝐴𝐴 = 𝐵)))
2322com23 86 . . . . 5 (𝐵 ∈ V → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2412, 23jaao 954 . . . 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 7743 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27 sucexb 7743 . . . . . 6 (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
2827notbii 320 . . . . 5 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29 nelneq 2858 . . . . 5 ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3026, 28, 29syl2anb 599 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3130pm2.21d 121 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
32 eqcom 2740 . . . 4 (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
3326notbii 320 . . . . . . 7 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34 nelneq 2858 . . . . . . 7 ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3527, 33, 34syl2anb 599 . . . . . 6 ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3635ancoms 460 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3736pm2.21d 121 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴𝐴 = 𝐵))
3832, 37biimtrid 241 . . 3 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
39 sucprc 6397 . . . . 5 𝐴 ∈ V → suc 𝐴 = 𝐴)
40 sucprc 6397 . . . . 5 𝐵 ∈ V → suc 𝐵 = 𝐵)
4139, 40eqeqan12d 2747 . . . 4 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4241biimpd 228 . . 3 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4325, 31, 38, 424cases 1040 . 2 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
44 suceq 6387 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4543, 44impbii 208 1 (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  Vcvv 3447  suc csuc 6323
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-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-reg 9536
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-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-eprel 5541  df-fr 5592  df-suc 6327
This theorem is referenced by:  rankxpsuc  9826  unidifsnel  31512  unidifsnne  31513  bnj551  33418  suceqsneq  36744  clsk1indlem1  42409
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