| Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > iotain | Structured version Visualization version GIF version | ||
| Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.) |
| Ref | Expression |
|---|---|
| iotain | ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eu6 2574 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 3 | 2 | intsn 4984 | . . . 4 ⊢ ∩ {𝑦} = 𝑦 |
| 4 | abbi 2807 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
| 5 | df-sn 4627 | . . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
| 7 | 6 | inteqd 4951 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦}) |
| 8 | iotaval 6532 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
| 9 | 3, 7, 8 | 3eqtr4a 2803 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
| 10 | 9 | exlimiv 1930 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
| 11 | 1, 10 | sylbi 217 | 1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∃wex 1779 ∃!weu 2568 {cab 2714 {csn 4626 ∩ cint 4946 ℩cio 6512 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-iota 6514 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |