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Mirrors > Home > MPE Home > Th. List > iotan0 | Structured version Visualization version GIF version |
Description: Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.) |
Ref | Expression |
---|---|
iotan0.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
iotan0 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm13.18 3020 | . . . . . 6 ⊢ ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅) | |
2 | 1 | expcom 413 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅)) |
3 | iotanul 6541 | . . . . . 6 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
4 | 3 | necon1ai 2966 | . . . . 5 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑) |
5 | 2, 4 | syl6 35 | . . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑)) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑))) |
7 | 6 | 3imp 1110 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → ∃!𝑥𝜑) |
8 | eqcom 2742 | . . . . 5 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴) | |
9 | iotan0.1 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
10 | 9 | iota2 6552 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
11 | 10 | biimprd 248 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴 → 𝜓)) |
12 | 8, 11 | biimtrid 242 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝐴 = (℩𝑥𝜑) → 𝜓)) |
13 | 12 | impancom 451 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓)) |
14 | 13 | 3adant2 1130 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓)) |
15 | 7, 14 | mpd 15 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 ≠ wne 2938 ∅c0 4339 ℩cio 6514 |
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 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-uni 4913 df-iota 6516 |
This theorem is referenced by: sgrpidmnd 18765 |
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