Theorem oppcmndclem 49049

Description: Lemma for oppcmndc 49051. 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 49050 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 2929	. 2 ⊢ (𝑋 ≠ 𝑌 ↔ ¬ 𝑋 = 𝑌)
2		eqeq1 2735	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
3		eqeq2 2743	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
4		oppcmndclem.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = {𝐴})
5		mosn 48844	. . . . . . 7 ⊢ (𝐵 = {𝐴} → ∃*𝑥 𝑥 ∈ 𝐵)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
7		moel 3366	. . . . . 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 3587	. . 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 1541   ∈ wcel 2111  ∃*wmo 2533   ≠ wne 2928  ∀wral 3047  {csn 4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-v 3438  df-sbc 3737  df-dif 3900  df-nul 4279  df-sn 4572

This theorem is referenced by:  oppcmndc  49051