Theorem eqoreldifOLD 4226

Description: Obsolete proof of eqoreldif 4225 as of 23-Jul-2021. (Contributed by AV, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
eqoreldifOLD	⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))

Proof of Theorem eqoreldifOLD

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))
2	1	a1d 25	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
3		simprr 796	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → 𝐴 ∈ 𝐶)
4		elsni 4194	. . . . . . . . . 10 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
5	4	a1i 11	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∈ {𝐵} → 𝐴 = 𝐵))
6	5	con3d 148	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (¬ 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵}))
7	6	impcom 446	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → ¬ 𝐴 ∈ {𝐵})
8	3, 7	eldifd 3585	. . . . . 6 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → 𝐴 ∈ (𝐶 ∖ {𝐵}))
9	8	olcd 408	. . . . 5 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))
10	9	ex 450	. . . 4 ⊢ (¬ 𝐴 = 𝐵 → ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
11	2, 10	pm2.61i 176	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))
12	11	ex 450	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
13		eleq1 2689	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
14	13	biimprd 238	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
15		eldifi 3732	. . . . 5 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴 ∈ 𝐶)
16	15	a1d 25	. . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐵}) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
17	14, 16	jaoi 394	. . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
18	17	com12 32	. 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})) → 𝐴 ∈ 𝐶))
19	12, 18	impbid 202	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by: (None)