Theorem oppcmndclem 48994

Description: Lemma for oppcmndc 48996. Everything is true for two distinct elements in a singleton or an empty set (since it is impossible). Note that if this theorem and oppcendc 48995 are in ¬ 𝑥 = 𝑦 form, then both proofs should be one step shorter. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypothesis

Ref	Expression
oppcmndclem.1	⊢ (𝜑 → 𝐵 = {𝐴})

Assertion

Ref	Expression
oppcmndclem	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → 𝜓))

Proof of Theorem oppcmndclem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2927	. 2 ⊢ (𝑋 ≠ 𝑌 ↔ ¬ 𝑋 = 𝑌)
2		eqeq1 2734	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
3		eqeq2 2742	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
4		oppcmndclem.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = {𝐴})
5		mosn 48791	. . . . . . 7 ⊢ (𝐵 = {𝐴} → ∃*𝑥 𝑥 ∈ 𝐵)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
7		moel 3378	. . . . . 6 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
8	6, 7	sylib 218	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
10		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
11		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
12	2, 3, 9, 10, 11	rspc2dv 3606	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 = 𝑌)
13	12	pm2.24d 151	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (¬ 𝑋 = 𝑌 → 𝜓))
14	1, 13	biimtrid 242	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃*wmo 2532   ≠ wne 2926  ∀wral 3045  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-v 3452  df-sbc 3756  df-dif 3919  df-nul 4299  df-sn 4592

This theorem is referenced by:  oppcmndc  48996