|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > disjenex | Structured version Visualization version GIF version | ||
| Description: Existence version of disjen 9175. (Contributed by Mario Carneiro, 7-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| disjenex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
| 2 | snex 5435 | . . 3 ⊢ {𝒫 ∪ ran 𝐴} ∈ V | |
| 3 | xpexg 7771 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ {𝒫 ∪ ran 𝐴} ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V) | |
| 4 | 1, 2, 3 | sylancl 586 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V) | 
| 5 | disjen 9175 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)) | |
| 6 | ineq2 4213 | . . . 4 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝐴 ∩ 𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴}))) | |
| 7 | 6 | eqeq1d 2738 | . . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅)) | 
| 8 | breq1 5145 | . . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝑥 ≈ 𝐵 ↔ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)) | |
| 9 | 7, 8 | anbi12d 632 | . 2 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))) | 
| 10 | 4, 5, 9 | spcedv 3597 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ∅c0 4332 𝒫 cpw 4599 {csn 4625 ∪ cuni 4906 class class class wbr 5142 × cxp 5682 ran crn 5685 ≈ cen 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1st 8015 df-2nd 8016 df-en 8987 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |