Theorem el7g 4652

Description: Members of a set with seven elements. Lemma for usgrexmpl2nb0 48652 etc. (Contributed by AV, 9-Aug-2025.)

Assertion

Ref	Expression
el7g	⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷) ∨ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺)))))

Proof of Theorem el7g

Step	Hyp	Ref	Expression
1		elun 4109	. 2 ⊢ (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 ∈ {𝐴} ∨ 𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})))
2		elsng 4599	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝐴} ↔ 𝑋 = 𝐴))
3		elun 4109	. . . 4 ⊢ (𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺}) ↔ (𝑋 ∈ {𝐵, 𝐶, 𝐷} ∨ 𝑋 ∈ {𝐸, 𝐹, 𝐺}))
4		eltpg 4648	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷)))
5		eltpg 4648	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝐸, 𝐹, 𝐺} ↔ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺)))
6	4, 5	orbi12d 931	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∈ {𝐵, 𝐶, 𝐷} ∨ 𝑋 ∈ {𝐸, 𝐹, 𝐺}) ↔ ((𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷) ∨ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺))))
7	3, 6	bitrid 286	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺}) ↔ ((𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷) ∨ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺))))
8	2, 7	orbi12d 931	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∈ {𝐴} ∨ 𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷) ∨ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺)))))
9	1, 8	bitrid 286	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷) ∨ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺)))))

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 860   ∨ w3o 1100   = wceq 1563   ∈ wcel 2145   ∪ cun 3905  {csn 4585  {ctp 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-sn 4586  df-pr 4588  df-tp 4590

This theorem is referenced by:  usgrexmpl2nb0  48652  usgrexmpl2nb1  48653  usgrexmpl2nb2  48654  usgrexmpl2nb3  48655  usgrexmpl2nb4  48656  usgrexmpl2nb5  48657