|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 3021 | . . . . . 6 ⊢ ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅) | |
| 2 | 1 | expcom 413 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅)) | 
| 3 | iotanul 6538 | . . . . . 6 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
| 4 | 3 | necon1ai 2967 | . . . . 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 2743 | . . . . 5 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴) | |
| 9 | iotan0.1 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 10 | 9 | iota2 6549 | . . . . . 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 1131 | . 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 1539 ∈ wcel 2107 ∃!weu 2567 ≠ wne 2939 ∅c0 4332 ℩cio 6511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 | 
| This theorem is referenced by: sgrpidmnd 18753 | 
| Copyright terms: Public domain | W3C validator |