Theorem oadif1lem 42114

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) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦)
oadif1lem.ord	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (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 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)})

Distinct variable groups:   𝐴,𝑏,𝑥,𝑦   𝐵,𝑏,𝑥,𝑦   ⊕ ,𝑏,𝑥,𝑦

Proof of Theorem oadif1lem

Step	Hyp	Ref	Expression
1		simpl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
2		oadif1lem.cl1	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊕ 𝐵) ∈ On)
3		onelon 6386	. . . . . . . . . . 11 ⊢ (((𝐴 ⊕ 𝐵) ∈ On ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵)) → 𝑦 ∈ On)
4	2, 3	sylan 580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵)) → 𝑦 ∈ On)
5		ontri1 6395	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
6	1, 4, 5	syl2an2r 683	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵)) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
7	6	pm5.32da 579	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ 𝐴 ⊆ 𝑦) ↔ (𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)))
8		ancom 461	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ 𝐴 ⊆ 𝑦) ↔ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵)))
9	7, 8	bitr3di 285	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵))))
10		oadif1lem.sub	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦)
11	9, 10	sylbida 592	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦)
12		eqcom 2739	. . . . . . 7 ⊢ ((𝐴 ⊕ 𝑏) = 𝑦 ↔ 𝑦 = (𝐴 ⊕ 𝑏))
13	12	rexbii 3094	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏))
14	11, 13	sylib 217	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏))
15	14	ex 413	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏)))
16		simpr 485	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 ⊕ 𝑏)) → 𝑦 = (𝐴 ⊕ 𝑏))
17		oadif1lem.ord	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 ⊕ 𝑏) ∈ (𝐴 ⊕ 𝐵)))
18	17	imp 407	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 ⊕ 𝑏) ∈ (𝐴 ⊕ 𝐵))
19	18	adantr 481	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 ⊕ 𝑏)) → (𝐴 ⊕ 𝑏) ∈ (𝐴 ⊕ 𝐵))
20	16, 19	eqeltrd 2833	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 ⊕ 𝑏)) → 𝑦 ∈ (𝐴 ⊕ 𝐵))
21		simpr 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
22		onelon 6386	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
23	21, 22	sylan 580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
24		oadif1lem.word	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 ⊕ 𝑏))
25	1, 23, 24	syl2an2r 683	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ (𝐴 ⊕ 𝑏))
26		oadif1lem.cl2	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 ⊕ 𝑏) ∈ On)
27	1, 23, 26	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 ⊕ 𝑏) ∈ On)
28		ontri1 6395	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐴 ⊕ 𝑏) ∈ On) → (𝐴 ⊆ (𝐴 ⊕ 𝑏) ↔ ¬ (𝐴 ⊕ 𝑏) ∈ 𝐴))
29	1, 27, 28	syl2an2r 683	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 ⊆ (𝐴 ⊕ 𝑏) ↔ ¬ (𝐴 ⊕ 𝑏) ∈ 𝐴))
30	25, 29	mpbid 231	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) → ¬ (𝐴 ⊕ 𝑏) ∈ 𝐴)
31	30	adantr 481	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 ⊕ 𝑏)) → ¬ (𝐴 ⊕ 𝑏) ∈ 𝐴)
32	16, 31	eqneltrd 2853	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 ⊕ 𝑏)) → ¬ 𝑦 ∈ 𝐴)
33	20, 32	jca 512	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 = (𝐴 ⊕ 𝑏)) → (𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
34	33	rexlimdva2 3157	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏) → (𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴)))
35	15, 34	impbid 211	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏)))
36		eldif 3957	. . 3 ⊢ (𝑦 ∈ ((𝐴 ⊕ 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 ⊕ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
37		vex 3478	. . . 4 ⊢ 𝑦 ∈ V
38		eqeq1 2736	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 ⊕ 𝑏) ↔ 𝑦 = (𝐴 ⊕ 𝑏)))
39	38	rexbidv 3178	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏)))
40	37, 39	elab 3667	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)} ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 ⊕ 𝑏))
41	35, 36, 40	3bitr4g 313	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ ((𝐴 ⊕ 𝐵) ∖ 𝐴) ↔ 𝑦 ∈ {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)}))
42	41	eqrdv 2730	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ⊕ 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070   ∖ cdif 3944   ⊆ wss 3947  Oncon0 6361  (class class class)co 7405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365

This theorem is referenced by: (None)