Theorem oadif1 43369

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 ∣ ∃𝑏 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
2		oacl 8571	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3		onelon 6410	. . . . . . . . . . 11 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → 𝑦 ∈ On)
4	2, 3	sylan 580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → 𝑦 ∈ On)
5		ontri1 6419	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
6	1, 4, 5	syl2an2r 685	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
7	6	pm5.32da 579	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝑦) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)))
8		ancom 460	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝑦) ↔ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 +o 𝐵)))
9	7, 8	bitr3di 286	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 +o 𝐵))))
10		oawordex2 43315	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 +o 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 +o 𝑏) = 𝑦)
11	9, 10	sylbida 592	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 (𝐴 +o 𝑏) = 𝑦)
12		eqcom 2741	. . . . . . 7 ⊢ ((𝐴 +o 𝑏) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑏))
13	12	rexbii 3091	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 (𝐴 +o 𝑏) = 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏))
14	11, 13	sylib 218	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏))
15	14	ex 412	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)))
16		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → 𝑦 = (𝐴 +o 𝑏))
17		oaordi 8582	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
18	17	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
19	18	imp 406	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
20	19	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
21	16, 20	eqeltrd 2838	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → 𝑦 ∈ (𝐴 +o 𝐵))
22		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
23		onelon 6410	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
24	22, 23	sylan 580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
25		oaword1 8588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑏))
26	1, 24, 25	syl2an2r 685	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ (𝐴 +o 𝑏))
27		oacl 8571	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 +o 𝑏) ∈ On)
28	1, 24, 27	syl2an2r 685	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +o 𝑏) ∈ On)
29		ontri1 6419	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐴 +o 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
30	1, 28, 29	syl2an2r 685	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
31	26, 30	mpbid 232	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → ¬ (𝐴 +o 𝑏) ∈ 𝐴)
32	31	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → ¬ (𝐴 +o 𝑏) ∈ 𝐴)
33	16, 32	eqneltrd 2858	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → ¬ 𝑦 ∈ 𝐴)
34	21, 33	jca 511	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
35	34	rexlimdva2 3154	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)))
36	15, 35	impbid 212	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)))
37		eldif 3972	. . 3 ⊢ (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
38		vex 3481	. . . 4 ⊢ 𝑦 ∈ V
39		eqeq1 2738	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 +o 𝑏) ↔ 𝑦 = (𝐴 +o 𝑏)))
40	39	rexbidv 3176	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)))
41	38, 40	elab 3680	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)} ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏))
42	36, 37, 41	3bitr4g 314	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ 𝑦 ∈ {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)}))
43	42	eqrdv 2732	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  {cab 2711  ∃wrex 3067   ∖ cdif 3959   ⊆ wss 3962  Oncon0 6385  (class class class)co 7430   +_o coa 8501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508

This theorem is referenced by:  oaun2  43370  oaun3  43371