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Theorem pr2ne 8772
Description: If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
Assertion
Ref Expression
pr2ne ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))

Proof of Theorem pr2ne
StepHypRef Expression
1 preq2 4239 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
21eqcoms 2629 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 enpr1g 7966 . . . . . . . 8 (𝐴𝐶 → {𝐴, 𝐴} ≈ 1𝑜)
4 entr 7952 . . . . . . . . . . . 12 (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1𝑜) → {𝐴, 𝐵} ≈ 1𝑜)
5 1sdom2 8103 . . . . . . . . . . . . . . 15 1𝑜 ≺ 2𝑜
6 sdomnen 7928 . . . . . . . . . . . . . . 15 (1𝑜 ≺ 2𝑜 → ¬ 1𝑜 ≈ 2𝑜)
75, 6ax-mp 5 . . . . . . . . . . . . . 14 ¬ 1𝑜 ≈ 2𝑜
8 ensym 7949 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ≈ 1𝑜 → 1𝑜 ≈ {𝐴, 𝐵})
9 entr 7952 . . . . . . . . . . . . . . . 16 ((1𝑜 ≈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
109ex 450 . . . . . . . . . . . . . . 15 (1𝑜 ≈ {𝐴, 𝐵} → ({𝐴, 𝐵} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
118, 10syl 17 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} ≈ 1𝑜 → ({𝐴, 𝐵} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
127, 11mtoi 190 . . . . . . . . . . . . 13 ({𝐴, 𝐵} ≈ 1𝑜 → ¬ {𝐴, 𝐵} ≈ 2𝑜)
1312a1d 25 . . . . . . . . . . . 12 ({𝐴, 𝐵} ≈ 1𝑜 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))
144, 13syl 17 . . . . . . . . . . 11 (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1𝑜) → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))
1514ex 450 . . . . . . . . . 10 ({𝐴, 𝐵} ≈ {𝐴, 𝐴} → ({𝐴, 𝐴} ≈ 1𝑜 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
16 prex 4870 . . . . . . . . . . 11 {𝐴, 𝐵} ∈ V
17 eqeng 7933 . . . . . . . . . . 11 ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} = {𝐴, 𝐴} → {𝐴, 𝐵} ≈ {𝐴, 𝐴}))
1816, 17ax-mp 5 . . . . . . . . . 10 ({𝐴, 𝐵} = {𝐴, 𝐴} → {𝐴, 𝐵} ≈ {𝐴, 𝐴})
1915, 18syl11 33 . . . . . . . . 9 ({𝐴, 𝐴} ≈ 1𝑜 → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
2019a1dd 50 . . . . . . . 8 ({𝐴, 𝐴} ≈ 1𝑜 → ({𝐴, 𝐵} = {𝐴, 𝐴} → (𝐵𝐷 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))))
213, 20syl 17 . . . . . . 7 (𝐴𝐶 → ({𝐴, 𝐵} = {𝐴, 𝐴} → (𝐵𝐷 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))))
2221com23 86 . . . . . 6 (𝐴𝐶 → (𝐵𝐷 → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))))
2322imp 445 . . . . 5 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
2423pm2.43a 54 . . . 4 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} = {𝐴, 𝐴} → ¬ {𝐴, 𝐵} ≈ 2𝑜))
252, 24syl5 34 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2𝑜))
2625necon2ad 2805 . 2 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))
27 pr2nelem 8771 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2𝑜)
28273expia 1264 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝐵 → {𝐴, 𝐵} ≈ 2𝑜))
2926, 28impbid 202 1 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  {cpr 4150   class class class wbr 4613  1𝑜c1o 7498  2𝑜c2o 7499  cen 7896  csdm 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-2o 7506  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  prdom2  8773  pmtrrn2  17801  mdetunilem7  20343
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