Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > reuun1 | Structured version Visualization version GIF version |
Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.) |
Ref | Expression |
---|---|
reuun1 | ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3738 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | orc 399 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 2 | rgenw 2908 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓)) |
4 | reuss2 3866 | . 2 ⊢ (((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓))) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓))) → ∃!𝑥 ∈ 𝐴 𝜑) | |
5 | 1, 3, 4 | mpanl12 714 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∀wral 2896 ∃wrex 2897 ∃!wreu 2898 ∪ cun 3538 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |