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Theorem oadif1lem 43392
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 482 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
2 oadif1lem.cl1 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 𝐵) ∈ On)
3 onelon 6409 . . . . . . . . . . 11 (((𝐴 𝐵) ∈ On ∧ 𝑦 ∈ (𝐴 𝐵)) → 𝑦 ∈ On)
42, 3sylan 580 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 𝐵)) → 𝑦 ∈ On)
5 ontri1 6418 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
61, 4, 5syl2an2r 685 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 𝐵)) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
76pm5.32da 579 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ 𝐴𝑦) ↔ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)))
8 ancom 460 . . . . . . . 8 ((𝑦 ∈ (𝐴 𝐵) ∧ 𝐴𝑦) ↔ (𝐴𝑦𝑦 ∈ (𝐴 𝐵)))
97, 8bitr3di 286 . . . . . . 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 592 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)) → ∃𝑏𝐵 (𝐴 𝑏) = 𝑦)
12 eqcom 2744 . . . . . . 7 ((𝐴 𝑏) = 𝑦𝑦 = (𝐴 𝑏))
1312rexbii 3094 . . . . . 6 (∃𝑏𝐵 (𝐴 𝑏) = 𝑦 ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏))
1411, 13sylib 218 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)) → ∃𝑏𝐵 𝑦 = (𝐴 𝑏))
1514ex 412 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑏𝐵 𝑦 = (𝐴 𝑏)))
16 simpr 484 . . . . . . 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 406 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 𝑏) ∈ (𝐴 𝐵))
1918adantr 480 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → (𝐴 𝑏) ∈ (𝐴 𝐵))
2016, 19eqeltrd 2841 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → 𝑦 ∈ (𝐴 𝐵))
21 simpr 484 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
22 onelon 6409 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
2321, 22sylan 580 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝑏 ∈ On)
24 oadif1lem.word . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 𝑏))
251, 23, 24syl2an2r 685 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝐴 ⊆ (𝐴 𝑏))
26 oadif1lem.cl2 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 𝑏) ∈ On)
271, 23, 26syl2an2r 685 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 𝑏) ∈ On)
28 ontri1 6418 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐴 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 𝑏) ↔ ¬ (𝐴 𝑏) ∈ 𝐴))
291, 27, 28syl2an2r 685 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 ⊆ (𝐴 𝑏) ↔ ¬ (𝐴 𝑏) ∈ 𝐴))
3025, 29mpbid 232 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → ¬ (𝐴 𝑏) ∈ 𝐴)
3130adantr 480 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → ¬ (𝐴 𝑏) ∈ 𝐴)
3216, 31eqneltrd 2861 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → ¬ 𝑦𝐴)
3320, 32jca 511 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 𝑏)) → (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴))
3433rexlimdva2 3157 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑏𝐵 𝑦 = (𝐴 𝑏) → (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴)))
3515, 34impbid 212 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴) ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏)))
36 eldif 3961 . . 3 (𝑦 ∈ ((𝐴 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 𝐵) ∧ ¬ 𝑦𝐴))
37 vex 3484 . . . 4 𝑦 ∈ V
38 eqeq1 2741 . . . . 5 (𝑥 = 𝑦 → (𝑥 = (𝐴 𝑏) ↔ 𝑦 = (𝐴 𝑏)))
3938rexbidv 3179 . . . 4 (𝑥 = 𝑦 → (∃𝑏𝐵 𝑥 = (𝐴 𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏)))
4037, 39elab 3679 . . 3 (𝑦 ∈ {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 𝑏)} ↔ ∃𝑏𝐵 𝑦 = (𝐴 𝑏))
4135, 36, 403bitr4g 314 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ ((𝐴 𝐵) ∖ 𝐴) ↔ 𝑦 ∈ {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 𝑏)}))
4241eqrdv 2735 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 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  cdif 3948  wss 3951  Oncon0 6384  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by: (None)
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