Theorem oppcmndclem 50094

Description: Lemma for oppcmndc 50096. 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 50095 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 2957	. 2 ⊢ (𝑋 ≠ 𝑌 ↔ ¬ 𝑋 = 𝑌)
2		eqeq1 2765	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
3		eqeq2 2773	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
4		oppcmndclem.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = {𝐴})
5		mosn 49892	. . . . . . 7 ⊢ (𝐵 = {𝐴} → ∃*𝑥 𝑥 ∈ 𝐵)
6	4, 5	syl 18	. . . . . 6 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
7		moel 3386	. . . . . 6 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
8	6, 7	sylib 221	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
9	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
10		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
11		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
12	2, 3, 9, 10, 11	rspc2dv 3591	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 = 𝑌)
13	12	pm2.24d 152	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (¬ 𝑋 = 𝑌 → 𝜓))
14	1, 13	biimtrid 245	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃*wmo 2563   ≠ wne 2956  ∀wral 3077  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-v 3453  df-sbc 3740  df-dif 3902  df-nul 4280  df-sn 4585

This theorem is used by:  oppcmndc  50096