Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unidifsnne Structured version   Visualization version   GIF version

Theorem unidifsnne 32562
Description: The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)
Assertion
Ref Expression
unidifsnne ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)

Proof of Theorem unidifsnne
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 2onn 8679 . . . . . . . . . 10 2o ∈ ω
2 nnfi 9206 . . . . . . . . . 10 (2o ∈ ω → 2o ∈ Fin)
31, 2ax-mp 5 . . . . . . . . 9 2o ∈ Fin
4 enfi 9225 . . . . . . . . 9 (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
53, 4mpbiri 258 . . . . . . . 8 (𝑃 ≈ 2o𝑃 ∈ Fin)
65adantl 481 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7 diffi 9214 . . . . . . 7 (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
86, 7syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
98cardidd 10587 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
109ensymd 9044 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11 simpl 482 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑋𝑃)
12 dif1card 10048 . . . . . . 7 ((𝑃 ∈ Fin ∧ 𝑋𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
136, 11, 12syl2anc 584 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
14 cardennn 10021 . . . . . . . . 9 ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
151, 14mpan2 691 . . . . . . . 8 (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16 df-2o 8506 . . . . . . . 8 2o = suc 1o
1715, 16eqtrdi 2791 . . . . . . 7 (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
1817adantl 481 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
1913, 18eqtr3d 2777 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20 suc11reg 9657 . . . . 5 (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
2119, 20sylib 218 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
2210, 21breqtrd 5174 . . 3 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23 en1 9063 . . 3 ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
2422, 23sylib 218 . 2 ((𝑋𝑃𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25 simplll 775 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋𝑃)
2625elexd 3502 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ V)
27 simplr 769 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑥})
28 sneqbg 4848 . . . . . . . . . . . . 13 (𝑋𝑃 → ({𝑋} = {𝑥} ↔ 𝑋 = 𝑥))
2928biimpar 477 . . . . . . . . . . . 12 ((𝑋𝑃𝑋 = 𝑥) → {𝑋} = {𝑥})
3029ad4ant14 752 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥})
3127, 30eqtr4d 2778 . . . . . . . . . 10 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋})
3231ineq2d 4228 . . . . . . . . 9 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋}))
33 disjdif 4478 . . . . . . . . 9 ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅
34 inidm 4235 . . . . . . . . 9 ({𝑋} ∩ {𝑋}) = {𝑋}
3532, 33, 343eqtr3g 2798 . . . . . . . 8 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋})
3635eqcomd 2741 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅)
37 snprc 4722 . . . . . . 7 𝑋 ∈ V ↔ {𝑋} = ∅)
3836, 37sylibr 234 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V)
3926, 38pm2.65da 817 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥)
4039neqned 2945 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋𝑥)
41 simpr 484 . . . . . 6 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
4241unieqd 4925 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
43 unisnv 4932 . . . . 5 {𝑥} = 𝑥
4442, 43eqtrdi 2791 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = 𝑥)
4540, 44neeqtrrd 3013 . . 3 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 (𝑃 ∖ {𝑋}))
4645necomd 2994 . 2 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
4724, 46exlimddv 1933 1 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  Vcvv 3478  cdif 3960  cin 3962  c0 4339  {csn 4631   cuni 4912   class class class wbr 5148  suc csuc 6388  cfv 6563  ωcom 7887  1oc1o 8498  2oc2o 8499  cen 8981  Fincfn 8984  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-ac 10154
This theorem is referenced by:  cyc3genpmlem  33154
  Copyright terms: Public domain W3C validator