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Theorem oadif1lem 42873
Description: Express the set difference of a continuous sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
Hypotheses
Ref Expression
oadif1lem.cl1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 𝐵) ∈ On)
oadif1lem.cl2 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 𝑏) ∈ On)
oadif1lem.sub (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝑦𝑦 ∈ (𝐴 𝐵))) → ∃𝑏𝐵 (𝐴 𝑏) = 𝑦)
oadif1lem.ord ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏𝐵 → (𝐴 𝑏) ∈ (𝐴 𝐵)))
oadif1lem.word ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 𝑏))
Assertion
Ref Expression
oadif1lem ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 𝑏)})
Distinct variable groups:   𝐴,𝑏,𝑥,𝑦   𝐵,𝑏,𝑥,𝑦   ,𝑏,𝑥,𝑦

Proof of Theorem oadif1lem
StepHypRef Expression
1 simpl 481 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
2 oadif1lem.cl1 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 𝐵) ∈ On)
3 onelon 6389 . . . . . . . . . . 11 (((𝐴 𝐵) ∈ On ∧ 𝑦 ∈ (𝐴 𝐵)) → 𝑦 ∈ On)
42, 3sylan 578 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 𝐵)) → 𝑦 ∈ On)
5 ontri1 6398 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
61, 4, 5syl2an2r 683 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 𝐵)) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
76pm5.32da 577 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ 𝐴𝑦) ↔ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)))
8 ancom 459 . . . . . . . 8 ((𝑦 ∈ (𝐴 𝐵) ∧ 𝐴𝑦) ↔ (𝐴𝑦𝑦 ∈ (𝐴 𝐵)))
97, 8bitr3di 285 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴) ↔ (𝐴𝑦𝑦 ∈ (𝐴 𝐵))))
10 oadif1lem.sub . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝑦𝑦 ∈ (𝐴 𝐵))) → ∃𝑏𝐵 (𝐴 𝑏) = 𝑦)
119, 10sylbida 590 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)) → ∃𝑏𝐵 (𝐴 𝑏) = 𝑦)
12 eqcom 2732 . . . . . . 7 ((𝐴 𝑏) = 𝑦𝑦 = (𝐴 𝑏))
1312rexbii 3084 . . . . . 6 (∃𝑏𝐵 (𝐴 𝑏) = 𝑦 ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏))
1411, 13sylib 217 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)) → ∃𝑏𝐵 𝑦 = (𝐴 𝑏))
1514ex 411 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑏𝐵 𝑦 = (𝐴 𝑏)))
16 simpr 483 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → 𝑦 = (𝐴 𝑏))
17 oadif1lem.ord . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏𝐵 → (𝐴 𝑏) ∈ (𝐴 𝐵)))
1817imp 405 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 𝑏) ∈ (𝐴 𝐵))
1918adantr 479 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → (𝐴 𝑏) ∈ (𝐴 𝐵))
2016, 19eqeltrd 2825 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → 𝑦 ∈ (𝐴 𝐵))
21 simpr 483 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
22 onelon 6389 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
2321, 22sylan 578 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝑏 ∈ On)
24 oadif1lem.word . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 𝑏))
251, 23, 24syl2an2r 683 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝐴 ⊆ (𝐴 𝑏))
26 oadif1lem.cl2 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 𝑏) ∈ On)
271, 23, 26syl2an2r 683 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 𝑏) ∈ On)
28 ontri1 6398 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐴 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 𝑏) ↔ ¬ (𝐴 𝑏) ∈ 𝐴))
291, 27, 28syl2an2r 683 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 ⊆ (𝐴 𝑏) ↔ ¬ (𝐴 𝑏) ∈ 𝐴))
3025, 29mpbid 231 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → ¬ (𝐴 𝑏) ∈ 𝐴)
3130adantr 479 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → ¬ (𝐴 𝑏) ∈ 𝐴)
3216, 31eqneltrd 2845 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → ¬ 𝑦𝐴)
3320, 32jca 510 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴))
3433rexlimdva2 3147 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑏𝐵 𝑦 = (𝐴 𝑏) → (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)))
3515, 34impbid 211 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴) ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏)))
36 eldif 3949 . . 3 (𝑦 ∈ ((𝐴 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴))
37 vex 3467 . . . 4 𝑦 ∈ V
38 eqeq1 2729 . . . . 5 (𝑥 = 𝑦 → (𝑥 = (𝐴 𝑏) ↔ 𝑦 = (𝐴 𝑏)))
3938rexbidv 3169 . . . 4 (𝑥 = 𝑦 → (∃𝑏𝐵 𝑥 = (𝐴 𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏)))
4037, 39elab 3659 . . 3 (𝑦 ∈ {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 𝑏)} ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏))
4135, 36, 403bitr4g 313 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ ((𝐴 𝐵) ∖ 𝐴) ↔ 𝑦 ∈ {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 𝑏)}))
4241eqrdv 2723 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 𝑏)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2702  wrex 3060  cdif 3936  wss 3939  Oncon0 6364  (class class class)co 7416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368
This theorem is referenced by: (None)
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