Theorem eqoreldifOLD 4173

Description: Obsolete proof of eqoreldif 4172 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 399	. . . . 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 792	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → 𝐴 ∈ 𝐶)
4		elsni 4142	. . . . . . . . . 10 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
5	4	a1i 11	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∈ {𝐵} → 𝐴 = 𝐵))
6	5	con3d 147	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (¬ 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵}))
7	6	impcom 445	. . . . . . 7 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → ¬ 𝐴 ∈ {𝐵})
8	3, 7	eldifd 3551	. . . . . 6 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → 𝐴 ∈ (𝐶 ∖ {𝐵}))
9	8	olcd 407	. . . . 5 ⊢ ((¬ 𝐴 = 𝐵 ∧ (𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶)) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))
10	9	ex 449	. . . 4 ⊢ (¬ 𝐴 = 𝐵 → ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
11	2, 10	pm2.61i 175	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))
12	11	ex 449	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
13		eleq1 2676	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
14	13	biimprd 237	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
15		eldifi 3694	. . . . 5 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴 ∈ 𝐶)
16	15	a1d 25	. . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐵}) → (𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶))
17	14, 16	jaoi 393	. . 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 201	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-sn 4126

This theorem is referenced by: (None)