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Mirrors > Home > ILE Home > Th. List > reuun1 | 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 3106 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | orc 633 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 2 | rgenw 2376 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓)) |
4 | reuss2 3217 | . 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 412 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∨ wo 629 ∀wral 2306 ∃wrex 2307 ∃!wreu 2308 ∪ cun 2915 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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