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Mirrors > Home > MPE Home > Th. List > morex | Structured version Visualization version GIF version |
Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
morex.1 | ⊢ 𝐵 ∈ V |
morex.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
morex | ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3072 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | exancom 1868 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | bitri 274 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) |
4 | nfmo1 2559 | . . . . . 6 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
5 | nfe1 2151 | . . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴) | |
6 | 4, 5 | nfan 1906 | . . . . 5 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) |
7 | mopick 2629 | . . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜑 → 𝑥 ∈ 𝐴)) | |
8 | 6, 7 | alrimi 2210 | . . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
9 | morex.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
10 | morex.2 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
11 | eleq1 2828 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
12 | 10, 11 | imbi12d 345 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜓 → 𝐵 ∈ 𝐴))) |
13 | 9, 12 | spcv 3543 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) → (𝜓 → 𝐵 ∈ 𝐴)) |
14 | 8, 13 | syl 17 | . . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜓 → 𝐵 ∈ 𝐴)) |
15 | 3, 14 | sylan2b 594 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
16 | 15 | ancoms 459 | 1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1540 = wceq 1542 ∃wex 1786 ∈ wcel 2110 ∃*wmo 2540 ∃wrex 3067 Vcvv 3431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-clab 2718 df-cleq 2732 df-clel 2818 df-rex 3072 df-v 3433 |
This theorem is referenced by: (None) |
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