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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 4145 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | orc 861 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 2 | rgenw 3147 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓)) |
4 | reuss2 4280 | . 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 698 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∀wral 3135 ∃wrex 3136 ∃!wreu 3137 ∪ cun 3931 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-reu 3142 df-v 3494 df-un 3938 df-in 3940 df-ss 3949 |
This theorem is referenced by: (None) |
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