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Theorem pr2ne 9038
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 4413 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
21eqcoms 2768 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 enpr1g 8189 . . . . . . . 8 (𝐴𝐶 → {𝐴, 𝐴} ≈ 1𝑜)
4 entr 8175 . . . . . . . . . . . 12 (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1𝑜) → {𝐴, 𝐵} ≈ 1𝑜)
5 1sdom2 8326 . . . . . . . . . . . . . . 15 1𝑜 ≺ 2𝑜
6 sdomnen 8152 . . . . . . . . . . . . . . 15 (1𝑜 ≺ 2𝑜 → ¬ 1𝑜 ≈ 2𝑜)
75, 6ax-mp 5 . . . . . . . . . . . . . 14 ¬ 1𝑜 ≈ 2𝑜
8 ensym 8172 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ≈ 1𝑜 → 1𝑜 ≈ {𝐴, 𝐵})
9 entr 8175 . . . . . . . . . . . . . . . 16 ((1𝑜 ≈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
109ex 449 . . . . . . . . . . . . . . 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 449 . . . . . . . . . 10 ({𝐴, 𝐵} ≈ {𝐴, 𝐴} → ({𝐴, 𝐴} ≈ 1𝑜 → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2𝑜)))
16 prex 5058 . . . . . . . . . . 11 {𝐴, 𝐵} ∈ V
17 eqeng 8157 . . . . . . . . . . 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 444 . . . . 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 2947 . 2 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))
27 pr2nelem 9037 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2𝑜)
28273expia 1115 . 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 383   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  {cpr 4323   class class class wbr 4804  1𝑜c1o 7723  2𝑜c2o 7724  cen 8120  csdm 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-1o 7730  df-2o 7731  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126
This theorem is referenced by:  prdom2  9039  isprm2lem  15616  pmtrrn2  18100  mdetunilem7  20646
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