Theorem djuexALT 10003

Description: Alternate proof of djuex 9989, 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 5395	. . 3 ⊢ {∅, 1o} ∈ V
2		unexg 7760	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
3		xpexg 7764	. . 3 ⊢ (({∅, 1o} ∈ V ∧ (𝐴 ∪ 𝐵) ∈ V) → ({∅, 1o} × (𝐴 ∪ 𝐵)) ∈ V)
4	1, 2, 3	sylancr 599	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({∅, 1o} × (𝐴 ∪ 𝐵)) ∈ V)
5		djuss 10001	. . 3 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)))
7	4, 6	ssexd 5285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {cpr 4585   × cxp 5648  1_oc1o 8469   ⊔ cdju 9979

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-suc 6368  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002  df-1o 8476  df-dju 9982  df-inl 9983  df-inr 9984

This theorem is used by: (None)