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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 4151 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | orc 863 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 2 | rgenw 3153 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓)) |
4 | reuss2 4286 | . 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 700 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∀wral 3141 ∃wrex 3142 ∃!wreu 3143 ∪ cun 3937 ⊆ wss 3939 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-reu 3148 df-v 3499 df-un 3944 df-in 3946 df-ss 3955 |
This theorem is referenced by: (None) |
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