Theorem el7g 4649

Description: Members of a set with seven elements. Lemma for usgrexmpl2nb0 48388 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 4107	. 2 ⊢ (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 ∈ {𝐴} ∨ 𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})))
2		elsng 4596	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝐴} ↔ 𝑋 = 𝐴))
3		elun 4107	. . . 4 ⊢ (𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺}) ↔ (𝑋 ∈ {𝐵, 𝐶, 𝐷} ∨ 𝑋 ∈ {𝐸, 𝐹, 𝐺}))
4		eltpg 4645	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷)))
5		eltpg 4645	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝐸, 𝐹, 𝐺} ↔ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺)))
6	4, 5	orbi12d 919	. . . 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 283	. . 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 919	. 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 283	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 206   ∨ wo 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114   ∪ cun 3901  {csn 4582  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  usgrexmpl2nb0  48388  usgrexmpl2nb1  48389  usgrexmpl2nb2  48390  usgrexmpl2nb3  48391  usgrexmpl2nb4  48392  usgrexmpl2nb5  48393