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Theorem pr2neOLD 9941
Description: Obsolete version of pr2ne 9940 as of 30-Dec-2024. (Contributed by FL, 14-Feb-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pr2neOLD ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Proof of Theorem pr2neOLD
StepHypRef Expression
1 preq2 4695 . . . . 5 (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
21eqcoms 2744 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴, 𝐴})
3 enpr1g 8964 . . . . . . . 8 (𝐴𝐶 → {𝐴, 𝐴} ≈ 1o)
4 entr 8946 . . . . . . . . . . . 12 (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1o) → {𝐴, 𝐵} ≈ 1o)
5 1sdom2 9184 . . . . . . . . . . . . . . 15 1o ≺ 2o
6 sdomnen 8921 . . . . . . . . . . . . . . 15 (1o ≺ 2o → ¬ 1o ≈ 2o)
75, 6ax-mp 5 . . . . . . . . . . . . . 14 ¬ 1o ≈ 2o
8 ensym 8943 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} ≈ 1o → 1o ≈ {𝐴, 𝐵})
9 entr 8946 . . . . . . . . . . . . . . . 16 ((1o ≈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ≈ 2o) → 1o ≈ 2o)
109ex 413 . . . . . . . . . . . . . . 15 (1o ≈ {𝐴, 𝐵} → ({𝐴, 𝐵} ≈ 2o → 1o ≈ 2o))
118, 10syl 17 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} ≈ 1o → ({𝐴, 𝐵} ≈ 2o → 1o ≈ 2o))
127, 11mtoi 198 . . . . . . . . . . . . 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 413 . . . . . . . . . 10 ({𝐴, 𝐵} ≈ {𝐴, 𝐴} → ({𝐴, 𝐴} ≈ 1o → ((𝐴𝐶𝐵𝐷) → ¬ {𝐴, 𝐵} ≈ 2o)))
16 prex 5389 . . . . . . . . . . 11 {𝐴, 𝐵} ∈ V
17 eqeng 8926 . . . . . . . . . . 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 407 . . . . 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 2958 . 2 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))
27 enpr2 9938 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)
28273expia 1121 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝐵 → {𝐴, 𝐵} ≈ 2o))
2926, 28impbid 211 1 ((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  {cpr 4588   class class class wbr 5105  1oc1o 8405  2oc2o 8406  cen 8880  csdm 8882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886
This theorem is referenced by: (None)
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