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Theorem el7g 4713
Description: Members of a set with seven elements. Lemma for usgrexmpl2nb0 47846 etc. (Contributed by AV, 9-Aug-2025.)
Assertion
Ref Expression
el7g (𝑋𝑉 → (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵𝑋 = 𝐶𝑋 = 𝐷) ∨ (𝑋 = 𝐸𝑋 = 𝐹𝑋 = 𝐺)))))

Proof of Theorem el7g
StepHypRef Expression
1 elun 4176 . 2 (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 ∈ {𝐴} ∨ 𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})))
2 elsng 4662 . . 3 (𝑋𝑉 → (𝑋 ∈ {𝐴} ↔ 𝑋 = 𝐴))
3 elun 4176 . . . 4 (𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺}) ↔ (𝑋 ∈ {𝐵, 𝐶, 𝐷} ∨ 𝑋 ∈ {𝐸, 𝐹, 𝐺}))
4 eltpg 4709 . . . . 5 (𝑋𝑉 → (𝑋 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝑋 = 𝐵𝑋 = 𝐶𝑋 = 𝐷)))
5 eltpg 4709 . . . . 5 (𝑋𝑉 → (𝑋 ∈ {𝐸, 𝐹, 𝐺} ↔ (𝑋 = 𝐸𝑋 = 𝐹𝑋 = 𝐺)))
64, 5orbi12d 917 . . . 4 (𝑋𝑉 → ((𝑋 ∈ {𝐵, 𝐶, 𝐷} ∨ 𝑋 ∈ {𝐸, 𝐹, 𝐺}) ↔ ((𝑋 = 𝐵𝑋 = 𝐶𝑋 = 𝐷) ∨ (𝑋 = 𝐸𝑋 = 𝐹𝑋 = 𝐺))))
73, 6bitrid 283 . . 3 (𝑋𝑉 → (𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺}) ↔ ((𝑋 = 𝐵𝑋 = 𝐶𝑋 = 𝐷) ∨ (𝑋 = 𝐸𝑋 = 𝐹𝑋 = 𝐺))))
82, 7orbi12d 917 . 2 (𝑋𝑉 → ((𝑋 ∈ {𝐴} ∨ 𝑋 ∈ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵𝑋 = 𝐶𝑋 = 𝐷) ∨ (𝑋 = 𝐸𝑋 = 𝐹𝑋 = 𝐺)))))
91, 8bitrid 283 1 (𝑋𝑉 → (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵𝑋 = 𝐶𝑋 = 𝐷) ∨ (𝑋 = 𝐸𝑋 = 𝐹𝑋 = 𝐺)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 846  w3o 1086   = wceq 1537  wcel 2108  cun 3974  {csn 4648  {ctp 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651  df-tp 4653
This theorem is referenced by:  usgrexmpl2nb0  47846  usgrexmpl2nb1  47847  usgrexmpl2nb2  47848  usgrexmpl2nb3  47849  usgrexmpl2nb4  47850  usgrexmpl2nb5  47851
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