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Theorem ordintdif 5887
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 4050 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 2943 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 dfdif2 3689 . . . 4 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
43inteqi 4587 . . 3 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
5 ordtri1 5869 . . . . . 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 5858 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
9 ordtri1 5869 . . . . . . . . . . 11 ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
107, 8, 9syl2anr 496 . . . . . . . . . 10 (((Ord 𝐴𝑥𝐴) ∧ Ord 𝐵) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1110an32s 881 . . . . . . . . 9 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1211rabbidva 3292 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
1312inteqd 4588 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
14 intmin 4605 . . . . . . 7 (𝐵𝐴 {𝑥𝐴𝐵𝑥} = 𝐵)
1513, 14sylan9req 2779 . . . . . 6 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵𝐴) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
1615ex 449 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
176, 16sylbird 250 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴𝐵 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
18173impia 1109 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
194, 18syl5req 2771 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → 𝐵 = (𝐴𝐵))
202, 19syl3an3br 1480 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  {crab 3018  cdif 3677  wss 3680  c0 4023   cint 4583  Ord word 5835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-ord 5839
This theorem is referenced by: (None)
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