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Theorem unidifsnne 30229
 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 8261 . . . . . . . . . 10 2o ∈ ω
2 nnfi 8705 . . . . . . . . . 10 (2o ∈ ω → 2o ∈ Fin)
31, 2ax-mp 5 . . . . . . . . 9 2o ∈ Fin
4 enfi 8728 . . . . . . . . 9 (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
53, 4mpbiri 259 . . . . . . . 8 (𝑃 ≈ 2o𝑃 ∈ Fin)
65adantl 482 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7 diffi 8744 . . . . . . 7 (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
86, 7syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
98cardidd 9965 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
109ensymd 8554 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11 simpl 483 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑋𝑃)
12 dif1card 9430 . . . . . . 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 9406 . . . . . . . . 9 ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
151, 14mpan2 687 . . . . . . . 8 (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16 df-2o 8099 . . . . . . . 8 2o = suc 1o
1715, 16syl6eq 2877 . . . . . . 7 (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
1817adantl 482 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
1913, 18eqtr3d 2863 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20 suc11reg 9076 . . . . 5 (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
2119, 20sylib 219 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
2210, 21breqtrd 5089 . . 3 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23 en1 8570 . . 3 ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
2422, 23sylib 219 . 2 ((𝑋𝑃𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25 simplll 771 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋𝑃)
2625elexd 3520 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ V)
27 simplr 765 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑥})
28 sneqbg 4773 . . . . . . . . . . . . 13 (𝑋𝑃 → ({𝑋} = {𝑥} ↔ 𝑋 = 𝑥))
2928biimpar 478 . . . . . . . . . . . 12 ((𝑋𝑃𝑋 = 𝑥) → {𝑋} = {𝑥})
3029ad4ant14 748 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥})
3127, 30eqtr4d 2864 . . . . . . . . . 10 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋})
3231ineq2d 4193 . . . . . . . . 9 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋}))
33 disjdif 4424 . . . . . . . . 9 ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅
34 inidm 4199 . . . . . . . . 9 ({𝑋} ∩ {𝑋}) = {𝑋}
3532, 33, 343eqtr3g 2884 . . . . . . . 8 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋})
3635eqcomd 2832 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅)
37 snprc 4652 . . . . . . 7 𝑋 ∈ V ↔ {𝑋} = ∅)
3836, 37sylibr 235 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V)
3926, 38pm2.65da 813 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥)
4039neqned 3028 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋𝑥)
41 simpr 485 . . . . . 6 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
4241unieqd 4847 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
43 vex 3503 . . . . . 6 𝑥 ∈ V
4443unisn 4853 . . . . 5 {𝑥} = 𝑥
4542, 44syl6eq 2877 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = 𝑥)
4640, 45neeqtrrd 3095 . . 3 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 (𝑃 ∖ {𝑋}))
4746necomd 3076 . 2 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
4824, 47exlimddv 1929 1 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107   ≠ wne 3021  Vcvv 3500   ∖ cdif 3937   ∩ cin 3939  ∅c0 4295  {csn 4564  ∪ cuni 4837   class class class wbr 5063  suc csuc 6192  ‘cfv 6354  ωcom 7573  1oc1o 8091  2oc2o 8092   ≈ cen 8500  Fincfn 8503  cardccrd 9358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-reg 9050  ax-ac2 9879 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-om 7574  df-wrecs 7943  df-recs 8004  df-1o 8098  df-2o 8099  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-ac 9536 This theorem is referenced by:  cyc3genpmlem  30726
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