Theorem djuexALT 9936

Description: Alternate proof of djuex 9922, 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 5407	. . 3 ⊢ {∅, 1o} ∈ V
2		unexg 7737	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
3		xpexg 7744	. . 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 9934	. . 3 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)))
7	4, 6	ssexd 5294	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  ∅c0 4308  {cpr 4603   × cxp 5652  1_oc1o 8473   ⊔ cdju 9912

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-suc 6358  df-iota 6484  df-fun 6533  df-fv 6539  df-1st 7988  df-2nd 7989  df-1o 8480  df-dju 9915  df-inl 9916  df-inr 9917

This theorem is referenced by: (None)