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Mathbox for Andrew Salmon |
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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 2569 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | vex 3451 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | intsn 4951 | . . . 4 ⊢ ∩ {𝑦} = 𝑦 |
4 | abbi 2801 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
5 | df-sn 4591 | . . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
6 | 4, 5 | eqtr4di 2791 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
7 | 6 | inteqd 4916 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦}) |
8 | iotaval 6471 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
9 | 3, 7, 8 | 3eqtr4a 2799 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
10 | 9 | exlimiv 1934 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
11 | 1, 10 | sylbi 216 | 1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∃!weu 2563 {cab 2710 {csn 4590 ∩ cint 4911 ℩cio 6450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-v 3449 df-un 3919 df-in 3921 df-ss 3931 df-sn 4591 df-pr 4593 df-uni 4870 df-int 4912 df-iota 6452 |
This theorem is referenced by: (None) |
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