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Theorem unidifsnne 30458
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 8297 . . . . . . . . . 10 2o ∈ ω
2 nnfi 8766 . . . . . . . . . 10 (2o ∈ ω → 2o ∈ Fin)
31, 2ax-mp 5 . . . . . . . . 9 2o ∈ Fin
4 enfi 8785 . . . . . . . . 9 (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
53, 4mpbiri 261 . . . . . . . 8 (𝑃 ≈ 2o𝑃 ∈ Fin)
65adantl 485 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7 diffi 8827 . . . . . . 7 (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
86, 7syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
98cardidd 10049 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
109ensymd 8606 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11 simpl 486 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2o) → 𝑋𝑃)
12 dif1card 9510 . . . . . . 7 ((𝑃 ∈ Fin ∧ 𝑋𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
136, 11, 12syl2anc 587 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
14 cardennn 9485 . . . . . . . . 9 ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
151, 14mpan2 691 . . . . . . . 8 (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16 df-2o 8132 . . . . . . . 8 2o = suc 1o
1715, 16eqtrdi 2789 . . . . . . 7 (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
1817adantl 485 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
1913, 18eqtr3d 2775 . . . . 5 ((𝑋𝑃𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20 suc11reg 9155 . . . . 5 (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
2119, 20sylib 221 . . . 4 ((𝑋𝑃𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
2210, 21breqtrd 5056 . . 3 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23 en1 8623 . . 3 ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
2422, 23sylib 221 . 2 ((𝑋𝑃𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25 simplll 775 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋𝑃)
2625elexd 3418 . . . . . 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 4729 . . . . . . . . . . . . 13 (𝑋𝑃 → ({𝑋} = {𝑥} ↔ 𝑋 = 𝑥))
2928biimpar 481 . . . . . . . . . . . 12 ((𝑋𝑃𝑋 = 𝑥) → {𝑋} = {𝑥})
3029ad4ant14 752 . . . . . . . . . . 11 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥})
3127, 30eqtr4d 2776 . . . . . . . . . 10 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋})
3231ineq2d 4103 . . . . . . . . 9 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋}))
33 disjdif 4361 . . . . . . . . 9 ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅
34 inidm 4109 . . . . . . . . 9 ({𝑋} ∩ {𝑋}) = {𝑋}
3532, 33, 343eqtr3g 2796 . . . . . . . 8 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋})
3635eqcomd 2744 . . . . . . 7 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅)
37 snprc 4608 . . . . . . 7 𝑋 ∈ V ↔ {𝑋} = ∅)
3836, 37sylibr 237 . . . . . 6 ((((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V)
3926, 38pm2.65da 817 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥)
4039neqned 2941 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋𝑥)
41 simpr 488 . . . . . 6 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
4241unieqd 4810 . . . . 5 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
43 vex 3402 . . . . . 6 𝑥 ∈ V
4443unisn 4818 . . . . 5 {𝑥} = 𝑥
4542, 44eqtrdi 2789 . . . 4 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = 𝑥)
4640, 45neeqtrrd 3008 . . 3 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 (𝑃 ∖ {𝑋}))
4746necomd 2989 . 2 (((𝑋𝑃𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
4824, 47exlimddv 1942 1 ((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  wne 2934  Vcvv 3398  cdif 3840  cin 3842  c0 4211  {csn 4516   cuni 4796   class class class wbr 5030  suc csuc 6174  cfv 6339  ωcom 7599  1oc1o 8124  2oc2o 8125  cen 8552  Fincfn 8555  cardccrd 9437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-reg 9129  ax-ac2 9963
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-om 7600  df-wrecs 7976  df-recs 8037  df-1o 8131  df-2o 8132  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-card 9441  df-ac 9616
This theorem is referenced by:  cyc3genpmlem  30995
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