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Theorem nnceleq 4430
 Description: If two naturals have an element in common, then they are equal. (Contributed by SF, 13-Feb-2015.)
Assertion
Ref Expression
nnceleq (((M Nn N Nn ) (A M A N)) → M = N)

Proof of Theorem nnceleq
StepHypRef Expression
1 elin 3219 . . . 4 (A (MN) ↔ (A M A N))
2 n0i 3555 . . . 4 (A (MN) → ¬ (MN) = )
31, 2sylbir 204 . . 3 ((A M A N) → ¬ (MN) = )
43adantl 452 . 2 (((M Nn N Nn ) (A M A N)) → ¬ (MN) = )
5 nndisjeq 4429 . . 3 ((M Nn N Nn ) → ((MN) = M = N))
65adantr 451 . 2 (((M Nn N Nn ) (A M A N)) → ((MN) = M = N))
7 orel1 371 . 2 (¬ (MN) = → (((MN) = M = N) → M = N))
84, 6, 7sylc 56 1 (((M Nn N Nn ) (A M A N)) → M = N)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  ∅c0 3550   Nn cnnc 4373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379 This theorem is referenced by:  prepeano4  4451  vfinnc  4471  ncfindi  4475  ncfinsn  4476  ncfineleq  4477  nnpw1ex  4484  tfin11  4493  tfinpw1  4494  ncfintfin  4495  tfindi  4496  tfin0c  4497  tfinsuc  4498  sfin112  4529  vfintle  4546  vfinspsslem1  4550  vfinncsp  4554
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