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Theorem ordintdif 5738
Description: If 𝐵 is smaller than 𝐴, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
Assertion
Ref Expression
ordintdif ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))

Proof of Theorem ordintdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssdif0 3921 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2837 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 dfdif2 3568 . . . 4 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
43inteqi 4449 . . 3 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
5 ordtri1 5720 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
65con2bid 344 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
7 id 22 . . . . . . . . . . 11 (Ord 𝐵 → Ord 𝐵)
8 ordelord 5709 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
9 ordtri1 5720 . . . . . . . . . . 11 ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
107, 8, 9syl2anr 495 . . . . . . . . . 10 (((Ord 𝐴𝑥𝐴) ∧ Ord 𝐵) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1110an32s 845 . . . . . . . . 9 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1211rabbidva 3179 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
1312inteqd 4450 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
14 intmin 4467 . . . . . . 7 (𝐵𝐴 {𝑥𝐴𝐵𝑥} = 𝐵)
1513, 14sylan9req 2676 . . . . . 6 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵𝐴) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
1615ex 450 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
176, 16sylbird 250 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴𝐵 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
18173impia 1258 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
194, 18syl5req 2668 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → 𝐵 = (𝐴𝐵))
202, 19syl3an3br 1364 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  cdif 3556  wss 3559  c0 3896   cint 4445  Ord word 5686
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
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-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690
This theorem is referenced by: (None)
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