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Theorem unidifsnne 32555
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 8681 . . . . . . . . . 10 2o ∈ ω
2 nnfi 9208 . . . . . . . . . 10 (2o ∈ ω → 2o ∈ Fin)
31, 2ax-mp 5 . . . . . . . . 9 2o ∈ Fin
4 enfi 9228 . . . . . . . . 9 (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
53, 4mpbiri 258 . . . . . . . 8 (𝑃 ≈ 2o𝑃 ∈ Fin)
65adantl 481 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7 diffi 9216 . . . . . . 7 (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
86, 7syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
98cardidd 10590 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
109ensymd 9046 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11 simpl 482 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑋𝑃)
12 dif1card 10051 . . . . . . 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 10024 . . . . . . . . 9 ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
151, 14mpan2 691 . . . . . . . 8 (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16 df-2o 8508 . . . . . . . 8 2o = suc 1o
1715, 16eqtrdi 2792 . . . . . . 7 (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
1817adantl 481 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
1913, 18eqtr3d 2778 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20 suc11reg 9660 . . . . 5 (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
2119, 20sylib 218 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
2210, 21breqtrd 5168 . . 3 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23 en1 9065 . . 3 ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
2422, 23sylib 218 . 2 ((𝑋𝑃𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25 simplll 774 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋𝑃)
2625elexd 3503 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ V)
27 simplr 768 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑥})
28 sneqbg 4842 . . . . . . . . . . . . 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 2779 . . . . . . . . . 10 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋})
3231ineq2d 4219 . . . . . . . . 9 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋}))
33 disjdif 4471 . . . . . . . . 9 ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅
34 inidm 4226 . . . . . . . . 9 ({𝑋} ∩ {𝑋}) = {𝑋}
3532, 33, 343eqtr3g 2799 . . . . . . . 8 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋})
3635eqcomd 2742 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅)
37 snprc 4716 . . . . . . 7 𝑋 ∈ V ↔ {𝑋} = ∅)
3836, 37sylibr 234 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V)
3926, 38pm2.65da 816 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥)
4039neqned 2946 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋𝑥)
41 simpr 484 . . . . . 6 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
4241unieqd 4919 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
43 unisnv 4926 . . . . 5 {𝑥} = 𝑥
4442, 43eqtrdi 2792 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = 𝑥)
4540, 44neeqtrrd 3014 . . 3 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 (𝑃 ∖ {𝑋}))
4645necomd 2995 . 2 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
4724, 46exlimddv 1934 1 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  wne 2939  Vcvv 3479  cdif 3947  cin 3949  c0 4332  {csn 4625   cuni 4906   class class class wbr 5142  suc csuc 6385  cfv 6560  ωcom 7888  1oc1o 8500  2oc2o 8501  cen 8983  Fincfn 8986  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-ac 10157
This theorem is referenced by:  cyc3genpmlem  33172
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