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Theorem unidifsnne 31638
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 8624 . . . . . . . . . 10 2o ∈ ω
2 nnfi 9150 . . . . . . . . . 10 (2o ∈ ω → 2o ∈ Fin)
31, 2ax-mp 5 . . . . . . . . 9 2o ∈ Fin
4 enfi 9173 . . . . . . . . 9 (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
53, 4mpbiri 257 . . . . . . . 8 (𝑃 ≈ 2o𝑃 ∈ Fin)
65adantl 482 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7 diffi 9162 . . . . . . 7 (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
86, 7syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
98cardidd 10526 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
109ensymd 8984 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11 simpl 483 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑋𝑃)
12 dif1card 9987 . . . . . . 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 9960 . . . . . . . . 9 ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
151, 14mpan2 689 . . . . . . . 8 (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16 df-2o 8449 . . . . . . . 8 2o = suc 1o
1715, 16eqtrdi 2787 . . . . . . 7 (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
1817adantl 482 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
1913, 18eqtr3d 2773 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20 suc11reg 9596 . . . . 5 (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
2119, 20sylib 217 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
2210, 21breqtrd 5167 . . 3 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23 en1 9004 . . 3 ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
2422, 23sylib 217 . 2 ((𝑋𝑃𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25 simplll 773 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋𝑃)
2625elexd 3493 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ V)
27 simplr 767 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑥})
28 sneqbg 4837 . . . . . . . . . . . . 13 (𝑋𝑃 → ({𝑋} = {𝑥} ↔ 𝑋 = 𝑥))
2928biimpar 478 . . . . . . . . . . . 12 ((𝑋𝑃𝑋 = 𝑥) → {𝑋} = {𝑥})
3029ad4ant14 750 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥})
3127, 30eqtr4d 2774 . . . . . . . . . 10 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋})
3231ineq2d 4208 . . . . . . . . 9 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋}))
33 disjdif 4467 . . . . . . . . 9 ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅
34 inidm 4214 . . . . . . . . 9 ({𝑋} ∩ {𝑋}) = {𝑋}
3532, 33, 343eqtr3g 2794 . . . . . . . 8 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋})
3635eqcomd 2737 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅)
37 snprc 4714 . . . . . . 7 𝑋 ∈ V ↔ {𝑋} = ∅)
3836, 37sylibr 233 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V)
3926, 38pm2.65da 815 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥)
4039neqned 2946 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋𝑥)
41 simpr 485 . . . . . 6 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
4241unieqd 4915 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
43 unisnv 4924 . . . . 5 {𝑥} = 𝑥
4442, 43eqtrdi 2787 . . . 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 1938 1 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2939  Vcvv 3473  cdif 3941  cin 3943  c0 4318  {csn 4622   cuni 4901   class class class wbr 5141  suc csuc 6355  cfv 6532  ωcom 7838  1oc1o 8441  2oc2o 8442  cen 8919  Fincfn 8922  cardccrd 9912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-reg 9569  ax-ac2 10440
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-1o 8448  df-2o 8449  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-card 9916  df-ac 10093
This theorem is referenced by:  cyc3genpmlem  32181
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