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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 2655 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | intsn 4904 | . . . 4 ⊢ ∩ {𝑦} = 𝑦 |
4 | abbi1 2884 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
5 | df-sn 4561 | . . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
6 | 4, 5 | syl6eqr 2874 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
7 | 6 | inteqd 4873 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦}) |
8 | iotaval 6323 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
9 | 3, 7, 8 | 3eqtr4a 2882 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
10 | 9 | exlimiv 1927 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
11 | 1, 10 | sylbi 219 | 1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∃wex 1776 ∃!weu 2649 {cab 2799 {csn 4560 ∩ cint 4868 ℩cio 6306 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-un 3940 df-in 3942 df-sn 4561 df-pr 4563 df-uni 4832 df-int 4869 df-iota 6308 |
This theorem is referenced by: (None) |
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