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Theorem pr2ne 8711
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 4213 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
21eqcoms 2618 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 enpr1g 7908 . . . . . . . 8 (𝐴𝐶 → {𝐴, 𝐴} ≈ 1𝑜)
4 prex 4836 . . . . . . . . . . . 12 {𝐴, 𝐵} ∈ V
5 eqeng 7875 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} = {𝐴, 𝐴} → {𝐴, 𝐵} ≈ {𝐴, 𝐴}))
64, 5ax-mp 5 . . . . . . . . . . 11 ({𝐴, 𝐵} = {𝐴, 𝐴} → {𝐴, 𝐵} ≈ {𝐴, 𝐴})
7 entr 7894 . . . . . . . . . . . . 13 (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1𝑜) → {𝐴, 𝐵} ≈ 1𝑜)
8 1sdom2 8044 . . . . . . . . . . . . . . . 16 1𝑜 ≺ 2𝑜
9 sdomnen 7870 . . . . . . . . . . . . . . . 16 (1𝑜 ≺ 2𝑜 → ¬ 1𝑜 ≈ 2𝑜)
108, 9ax-mp 5 . . . . . . . . . . . . . . 15 ¬ 1𝑜 ≈ 2𝑜
11 ensym 7891 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} ≈ 1𝑜 → 1𝑜 ≈ {𝐴, 𝐵})
12 entr 7894 . . . . . . . . . . . . . . . . 17 ((1𝑜 ≈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
1312ex 449 . . . . . . . . . . . . . . . 16 (1𝑜 ≈ {𝐴, 𝐵} → ({𝐴, 𝐵} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
1411, 13syl 17 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ≈ 1𝑜 → ({𝐴, 𝐵} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
1510, 14mtoi 189 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} ≈ 1𝑜 → ¬ {𝐴, 𝐵} ≈ 2𝑜)
1615a1d 25 . . . . . . . . . . . . 13 ({𝐴, 𝐵} ≈ 1𝑜 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))
177, 16syl 17 . . . . . . . . . . . 12 (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1𝑜) → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))
1817ex 449 . . . . . . . . . . 11 ({𝐴, 𝐵} ≈ {𝐴, 𝐴} → ({𝐴, 𝐴} ≈ 1𝑜 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
196, 18syl 17 . . . . . . . . . 10 ({𝐴, 𝐵} = {𝐴, 𝐴} → ({𝐴, 𝐴} ≈ 1𝑜 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
2019com12 32 . . . . . . . . 9 ({𝐴, 𝐴} ≈ 1𝑜 → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
2120a1dd 48 . . . . . . . 8 ({𝐴, 𝐴} ≈ 1𝑜 → ({𝐴, 𝐵} = {𝐴, 𝐴} → (𝐵𝐷 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))))
223, 21syl 17 . . . . . . 7 (𝐴𝐶 → ({𝐴, 𝐵} = {𝐴, 𝐴} → (𝐵𝐷 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))))
2322com23 84 . . . . . 6 (𝐴𝐶 → (𝐵𝐷 → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜))))
2423imp 444 . . . . 5 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
2524pm2.43a 52 . . . 4 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} = {𝐴, 𝐴} → ¬ {𝐴, 𝐵} ≈ 2𝑜))
262, 25syl5 33 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2𝑜))
2726necon2ad 2797 . 2 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))
28 pr2nelem 8710 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2𝑜)
29283expia 1259 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝐵 → {𝐴, 𝐵} ≈ 2𝑜))
3027, 29impbid 201 1 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  {cpr 4127   class class class wbr 4583  1𝑜c1o 7440  2𝑜c2o 7441  cen 7838  csdm 7840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844
This theorem is referenced by:  prdom2  8712  pmtrrn2  17703  mdetunilem7  20243
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