Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oadif1 Structured version   Visualization version   GIF version

Theorem oadif1 42687
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 8533 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3 onelon 6382 . . . . . . . . . . 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 6391 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
61, 4, 5syl2an2r 682 . . . . . . . . 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 42633 . . . . . . 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 2733 . . . . . . 7 ((𝐴 +o 𝑏) = 𝑦𝑦 = (𝐴 +o 𝑏))
1312rexbii 3088 . . . . . 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 8544 . . . . . . . . . 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 2827 . . . . . 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 6382 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
2422, 23sylan 579 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝑏 ∈ On)
25 oaword1 8550 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑏))
261, 24, 25syl2an2r 682 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → 𝐴 ⊆ (𝐴 +o 𝑏))
27 oacl 8533 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 +o 𝑏) ∈ On)
281, 24, 27syl2an2r 682 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏𝐵) → (𝐴 +o 𝑏) ∈ On)
29 ontri1 6391 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐴 +o 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
301, 28, 29syl2an2r 682 . . . . . . . . 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 2847 . . . . . 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 3151 . . . 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 3953 . . 3 (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
38 vex 3472 . . . 4 𝑦 ∈ V
39 eqeq1 2730 . . . . 5 (𝑥 = 𝑦 → (𝑥 = (𝐴 +o 𝑏) ↔ 𝑦 = (𝐴 +o 𝑏)))
4039rexbidv 3172 . . . 4 (𝑥 = 𝑦 → (∃𝑏𝐵 𝑥 = (𝐴 +o 𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝐴 +o 𝑏)))
4138, 40elab 3663 . . 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 2724 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 1533  wcel 2098  {cab 2703  wrex 3064  cdif 3940  wss 3943  Oncon0 6357  (class class class)co 7404   +o coa 8461
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-oadd 8468
This theorem is referenced by:  oaun2  42688  oaun3  42689
  Copyright terms: Public domain W3C validator