MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suc11reg Structured version   Visualization version   GIF version

Theorem suc11reg 8876
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 8864 . . . . 5 ¬ (𝐴𝐵𝐵𝐴)
2 ianor 964 . . . . 5 (¬ (𝐴𝐵𝐵𝐴) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐵𝐴))
31, 2mpbi 222 . . . 4 𝐴𝐵 ∨ ¬ 𝐵𝐴)
4 sucidg 6107 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5 eleq2 2854 . . . . . . . . . . 11 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
64, 5syl5ibcom 237 . . . . . . . . . 10 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
7 elsucg 6096 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
86, 7sylibd 231 . . . . . . . . 9 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
98imp 398 . . . . . . . 8 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
109ord 850 . . . . . . 7 ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴𝐵𝐴 = 𝐵))
1110ex 405 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵)))
1211com23 86 . . . . 5 (𝐴 ∈ V → (¬ 𝐴𝐵 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
13 sucidg 6107 . . . . . . . . . . . 12 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14 eleq2 2854 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
1513, 14syl5ibrcom 239 . . . . . . . . . . 11 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
16 elsucg 6096 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
1715, 16sylibd 231 . . . . . . . . . 10 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
1817imp 398 . . . . . . . . 9 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
1918ord 850 . . . . . . . 8 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐵 = 𝐴))
20 eqcom 2785 . . . . . . . 8 (𝐵 = 𝐴𝐴 = 𝐵)
2119, 20syl6ib 243 . . . . . . 7 ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵𝐴𝐴 = 𝐵))
2221ex 405 . . . . . 6 (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵𝐴𝐴 = 𝐵)))
2322com23 86 . . . . 5 (𝐵 ∈ V → (¬ 𝐵𝐴 → (suc 𝐴 = suc 𝐵𝐴 = 𝐵)))
2412, 23jaao 937 . . . 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 7340 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27 sucexb 7340 . . . . . 6 (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
2827notbii 312 . . . . 5 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29 nelneq 2890 . . . . 5 ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3026, 28, 29syl2anb 588 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
3130pm2.21d 119 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
32 eqcom 2785 . . . 4 (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
3326notbii 312 . . . . . . 7 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34 nelneq 2890 . . . . . . 7 ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3527, 33, 34syl2anb 588 . . . . . 6 ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3635ancoms 451 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
3736pm2.21d 119 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴𝐴 = 𝐵))
3832, 37syl5bi 234 . . 3 ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
39 sucprc 6104 . . . . 5 𝐴 ∈ V → suc 𝐴 = 𝐴)
40 sucprc 6104 . . . . 5 𝐵 ∈ V → suc 𝐵 = 𝐵)
4139, 40eqeqan12d 2794 . . . 4 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4241biimpd 221 . . 3 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
4325, 31, 38, 424cases 1021 . 2 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
44 suceq 6094 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4543, 44impbii 201 1 (suc 𝐴 = suc 𝐵𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833   = wceq 1507  wcel 2050  Vcvv 3415  suc csuc 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279  ax-reg 8851
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-eprel 5317  df-fr 5366  df-suc 6035
This theorem is referenced by:  rankxpsuc  9105  bnj551  31667  clsk1indlem1  39764
  Copyright terms: Public domain W3C validator