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Theorem ordintdif 5691
 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 3896 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2829 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 dfdif2 3549 . . . 4 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
43inteqi 4414 . . 3 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
5 ordtri1 5673 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
65con2bid 343 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
7 id 22 . . . . . . . . . . 11 (Ord 𝐵 → Ord 𝐵)
8 ordelord 5662 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
9 ordtri1 5673 . . . . . . . . . . 11 ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
107, 8, 9syl2anr 494 . . . . . . . . . 10 (((Ord 𝐴𝑥𝐴) ∧ Ord 𝐵) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1110an32s 842 . . . . . . . . 9 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1211rabbidva 3163 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
1312inteqd 4415 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
14 intmin 4432 . . . . . . 7 (𝐵𝐴 {𝑥𝐴𝐵𝑥} = 𝐵)
1513, 14sylan9req 2665 . . . . . 6 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵𝐴) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
1615ex 449 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
176, 16sylbird 249 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴𝐵 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
18173impia 1253 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
194, 18syl5req 2657 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → 𝐵 = (𝐴𝐵))
202, 19syl3an3br 1359 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Ord word 5639 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-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-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643 This theorem is referenced by: (None)
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