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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 3063 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑) |
3 | reuss2 4332 | . . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜑)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑) | |
4 | 2, 3 | mpanl2 701 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ∃!𝑥 ∈ 𝐴 𝜑) |
5 | 4 | 3impb 1114 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ex 1777 df-mo 2538 df-eu 2567 df-clel 2814 df-ral 3060 df-rex 3069 df-reu 3379 df-ss 3980 |
This theorem is referenced by: euelss 4338 riotass 7419 adjbdln 32112 tfsconcatlem 43326 tfsconcatfv 43331 |
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