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Theorem suc11reg 8678
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 8664 . . . . 5 ¬ (𝐴𝐵𝐵𝐴)
2 ianor 511 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
31, 2mpbi 220 . . . 4 𝐴𝐵 ∨ ¬ 𝐵𝐴)
4 sucidg 5945 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5 eleq2 2837 . . . . . . . . . . 11 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
64, 5syl5ibcom 235 . . . . . . . . . 10 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
7 elsucg 5934 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
86, 7sylibd 229 . . . . . . . . 9 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
98imp 443 . . . . . . . 8 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
109ord 391 . . . . . . 7 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴𝐵𝐴 = 𝐵))
1110ex 448 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵)))
1211com23 86 . . . . 5 (𝐴 ∈ V → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 sucidg 5945 . . . . . . . . . . . 12 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14 eleq2 2837 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
1513, 14syl5ibrcom 237 . . . . . . . . . . 11 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsucg 5934 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
1715, 16sylibd 229 . . . . . . . . . 10 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
1817imp 443 . . . . . . . . 9 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
1918ord 391 . . . . . . . 8 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐵 = 𝐴))
20 eqcom 2776 . . . . . . . 8 (𝐵 = 𝐴𝐴 = 𝐵)
2119, 20syl6ib 241 . . . . . . 7 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐴 = 𝐵))
2221ex 448 . . . . . 6 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵𝐴𝐴 = 𝐵)))
2322com23 86 . . . . 5 (𝐵 ∈ V → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2412, 23jaao 533 . . . 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 7154 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27 sucexb 7154 . . . . . 6 (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
2827notbii 309 . . . . 5 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29 nelneq 2872 . . . . 5 ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3026, 28, 29syl2anb 498 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3130pm2.21d 119 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
32 eqcom 2776 . . . 4 (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
3326notbii 309 . . . . . . 7 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34 nelneq 2872 . . . . . . 7 ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3527, 33, 34syl2anb 498 . . . . . 6 ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3635ancoms 469 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3736pm2.21d 119 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴𝐴 = 𝐵))
3832, 37syl5bi 232 . . 3 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
39 sucprc 5942 . . . . 5 𝐴 ∈ V → suc 𝐴 = 𝐴)
40 sucprc 5942 . . . . 5 𝐵 ∈ V → suc 𝐵 = 𝐵)
4139, 40eqeqan12d 2785 . . . 4 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4241biimpd 219 . . 3 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4325, 31, 38, 424cases 1024 . 2 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
44 suceq 5932 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4543, 44impbii 199 1 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1629  wcel 2143  Vcvv 3348  suc csuc 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033  ax-un 7094  ax-reg 8651
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-eprel 5161  df-fr 5207  df-suc 5871
This theorem is referenced by:  rankxpsuc  8907  bnj551  31151  clsk1indlem1  38869
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