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Theorem ordintdif 6218
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 4295 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
21necon3bbii 3058 . 2 𝐴𝐵 ↔ (𝐴𝐵) ≠ ∅)
3 dfdif2 3917 . . . 4 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
43inteqi 4855 . . 3 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
5 ordtri1 6202 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
65con2bid 358 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
7 id 22 . . . . . . . . . . 11 (Ord 𝐵 → Ord 𝐵)
8 ordelord 6191 . . . . . . . . . . 11 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
9 ordtri1 6202 . . . . . . . . . . 11 ((Ord 𝐵 ∧ Ord 𝑥) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
107, 8, 9syl2anr 599 . . . . . . . . . 10 (((Ord 𝐴𝑥𝐴) ∧ Ord 𝐵) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1110an32s 651 . . . . . . . . 9 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝐵𝑥 ↔ ¬ 𝑥𝐵))
1211rabbidva 3453 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
1312inteqd 4856 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → {𝑥𝐴𝐵𝑥} = {𝑥𝐴 ∣ ¬ 𝑥𝐵})
14 intmin 4871 . . . . . . 7 (𝐵𝐴 {𝑥𝐴𝐵𝑥} = 𝐵)
1513, 14sylan9req 2878 . . . . . 6 (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐵𝐴) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
1615ex 416 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
176, 16sylbird 263 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴𝐵 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵))
18173impia 1114 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → {𝑥𝐴 ∣ ¬ 𝑥𝐵} = 𝐵)
194, 18syl5req 2870 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ ¬ 𝐴𝐵) → 𝐵 = (𝐴𝐵))
202, 19syl3an3br 1405 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  {crab 3134  cdif 3905  wss 3908  c0 4265   cint 4851  Ord word 6168
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172
This theorem is referenced by: (None)
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