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Theorem oadif1 42809
Description: Express the set difference of an ordinal sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
Assertion
Ref Expression
oadif1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 +o 𝑏)})
Distinct variable groups:   𝐴,𝑏,𝑥   𝐵,𝑏,𝑥

Proof of Theorem oadif1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
2 oacl 8556 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3 onelon 6394 . . . . . . . . . . 11 (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → 𝑦 ∈ On)
42, 3sylan 579 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → 𝑦 ∈ On)
5 ontri1 6403 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
61, 4, 5syl2an2r 684 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
76pm5.32da 578 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ 𝐴𝑦) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴)))
8 ancom 460 . . . . . . . 8 ((𝑦 ∈ (𝐴 +o 𝐵) ∧ 𝐴𝑦) ↔ (𝐴𝑦𝑦 ∈ (𝐴 +o 𝐵)))
97, 8bitr3di 286 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴) ↔ (𝐴𝑦𝑦 ∈ (𝐴 +o 𝐵))))
10 oawordex2 42755 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝑦𝑦 ∈ (𝐴 +o 𝐵))) → ∃𝑏𝐵 (𝐴 +o 𝑏) = 𝑦)
119, 10sylbida 591 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴)) → ∃𝑏𝐵 (𝐴 +o 𝑏) = 𝑦)
12 eqcom 2735 . . . . . . 7 ((𝐴 +o 𝑏) = 𝑦𝑦 = (𝐴 +o 𝑏))
1312rexbii 3091 . . . . . 6 (∃𝑏𝐵 (𝐴 +o 𝑏) = 𝑦 ↔ ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏))
1411, 13sylib 217 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴)) → ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏))
1514ex 412 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏)))
16 simpr 484 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → 𝑦 = (𝐴 +o 𝑏))
17 oaordi 8567 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑏𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
1817ancoms 458 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
1918imp 406 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
2019adantr 480 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
2116, 20eqeltrd 2829 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → 𝑦 ∈ (𝐴 +o 𝐵))
22 simpr 484 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
23 onelon 6394 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
2422, 23sylan 579 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝑏 ∈ On)
25 oaword1 8573 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑏))
261, 24, 25syl2an2r 684 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝐴 ⊆ (𝐴 +o 𝑏))
27 oacl 8556 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 +o 𝑏) ∈ On)
281, 24, 27syl2an2r 684 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 +o 𝑏) ∈ On)
29 ontri1 6403 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐴 +o 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
301, 28, 29syl2an2r 684 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
3126, 30mpbid 231 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → ¬ (𝐴 +o 𝑏) ∈ 𝐴)
3231adantr 480 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → ¬ (𝐴 +o 𝑏) ∈ 𝐴)
3316, 32eqneltrd 2849 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → ¬ 𝑦𝐴)
3421, 33jca 511 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
3534rexlimdva2 3154 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴)))
3615, 35impbid 211 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴) ↔ ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏)))
37 eldif 3957 . . 3 (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
38 vex 3475 . . . 4 𝑦 ∈ V
39 eqeq1 2732 . . . . 5 (𝑥 = 𝑦 → (𝑥 = (𝐴 +o 𝑏) ↔ 𝑦 = (𝐴 +o 𝑏)))
4039rexbidv 3175 . . . 4 (𝑥 = 𝑦 → (∃𝑏𝐵 𝑥 = (𝐴 +o 𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏)))
4138, 40elab 3667 . . 3 (𝑦 ∈ {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 +o 𝑏)} ↔ ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏))
4236, 37, 413bitr4g 314 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ 𝑦 ∈ {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 +o 𝑏)}))
4342eqrdv 2726 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏𝐵 𝑥 = (𝐴 +o 𝑏)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  cdif 3944  wss 3947  Oncon0 6369  (class class class)co 7420   +o coa 8484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-oadd 8491
This theorem is referenced by:  oaun2  42810  oaun3  42811
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