Theorem oadif1 43962

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 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
2		oacl 8504	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3		onelon 6371	. . . . . . . . . . 11 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → 𝑦 ∈ On)
4	2, 3	sylan 589	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → 𝑦 ∈ On)
5		ontri1 6380	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
6	1, 4, 5	syl2an2r 695	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 +o 𝐵)) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
7	6	pm5.32da 587	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝑦) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)))
8		ancom 464	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ 𝐴 ⊆ 𝑦) ↔ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 +o 𝐵)))
9	7, 8	bitr3di 288	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 +o 𝐵))))
10		oawordex2 43908	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 +o 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 +o 𝑏) = 𝑦)
11	9, 10	sylbida 601	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 (𝐴 +o 𝑏) = 𝑦)
12		eqcom 2770	. . . . . . 7 ⊢ ((𝐴 +o 𝑏) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑏))
13	12	rexbii 3110	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 (𝐴 +o 𝑏) = 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏))
14	11, 13	sylib 220	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏))
15	14	ex 416	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)))
16		simpr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → 𝑦 = (𝐴 +o 𝑏))
17		oaordi 8515	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
18	17	ancoms 462	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
19	18	imp 410	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
20	19	adantr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
21	16, 20	eqeltrd 2863	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → 𝑦 ∈ (𝐴 +o 𝐵))
22		simpr 488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
23		onelon 6371	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
24	22, 23	sylan 589	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
25		oaword1 8521	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑏))
26	1, 24, 25	syl2an2r 695	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ (𝐴 +o 𝑏))
27		oacl 8504	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 +o 𝑏) ∈ On)
28	1, 24, 27	syl2an2r 695	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +o 𝑏) ∈ On)
29		ontri1 6380	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐴 +o 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
30	1, 28, 29	syl2an2r 695	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 ⊆ (𝐴 +o 𝑏) ↔ ¬ (𝐴 +o 𝑏) ∈ 𝐴))
31	26, 30	mpbid 234	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → ¬ (𝐴 +o 𝑏) ∈ 𝐴)
32	31	adantr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → ¬ (𝐴 +o 𝑏) ∈ 𝐴)
33	16, 32	eqneltrd 2883	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → ¬ 𝑦 ∈ 𝐴)
34	21, 33	jca 519	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 +o 𝑏)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
35	34	rexlimdva2 3166	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)))
36	15, 35	impbid 214	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)))
37		eldif 3915	. . 3 ⊢ (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
38		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
39		eqeq1 2767	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 +o 𝑏) ↔ 𝑦 = (𝐴 +o 𝑏)))
40	39	rexbidv 3187	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)))
41	38, 40	elab 3639	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)} ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏))
42	36, 37, 41	3bitr4g 316	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ 𝑦 ∈ {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)}))
43	42	eqrdv 2761	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {cab 2741  ∃wrex 3087   ∖ cdif 3902   ⊆ wss 3905  Oncon0 6346  (class class class)co 7396   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-oadd 8441

This theorem is referenced by:  oaun2  43963  oaun3  43964