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Mirrors > Home > MPE Home > Th. List > reuss | Structured version Visualization version GIF version |
Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.) |
Ref | Expression |
---|---|
reuss | ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝜑 → 𝜑) | |
2 | 1 | rgenw 2924 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑) |
3 | reuss2 3907 | . . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑) | |
4 | 2, 3 | mpanl2 717 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑) |
5 | 4 | 3impb 1260 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ∀wral 2912 ∃wrex 2913 ∃!wreu 2914 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-ral 2917 df-rex 2918 df-reu 2919 df-in 3581 df-ss 3588 |
This theorem is referenced by: euelss 3914 riotass 6639 adjbdln 28942 |
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