Theorem djuexALT 9818

Description: Alternate proof of djuex 9804, which is shorter, but based indirectly on the definitions of inl and inr. (Proposed by BJ, 28-Jun-2022.) (Contributed by AV, 28-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
djuexALT	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)

Proof of Theorem djuexALT

Step	Hyp	Ref	Expression
1		prex 5376	. . 3 ⊢ {∅, 1o} ∈ V
2		unexg 7679	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
3		xpexg 7686	. . 3 ⊢ (({∅, 1o} ∈ V ∧ (𝐴 ∪ 𝐵) ∈ V) → ({∅, 1o} × (𝐴 ∪ 𝐵)) ∈ V)
4	1, 2, 3	sylancr 587	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅, 1o} × (𝐴 ∪ 𝐵)) ∈ V)
5		djuss 9816	. . 3 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)))
7	4, 6	ssexd 5263	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {cpr 4579   × cxp 5617  1_oc1o 8381   ⊔ cdju 9794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-suc 6313  df-iota 6438  df-fun 6484  df-fv 6490  df-1st 7924  df-2nd 7925  df-1o 8388  df-dju 9797  df-inl 9798  df-inr 9799

This theorem is referenced by: (None)