| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > unexb | Structured version Visualization version GIF version | ||
| Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
| Ref | Expression |
|---|---|
| unexb | ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unexg 7763 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
| 2 | ssun1 4178 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
| 3 | ssexg 5323 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) | |
| 4 | 2, 3 | mpan 690 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V) |
| 5 | ssun2 4179 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
| 6 | ssexg 5323 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) | |
| 7 | 5, 6 | mpan 690 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V) |
| 8 | 4, 7 | jca 511 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| 9 | 1, 8 | impbii 209 | 1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: unexgOLD 7769 sucexb 7824 fodomr 9168 fsuppun 9427 fsuppunbi 9429 djuexb 9949 bj-tagex 36988 |
| Copyright terms: Public domain | W3C validator |