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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 2567 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | vex 3477 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | intsn 4983 | . . . 4 ⊢ ∩ {𝑦} = 𝑦 |
4 | abbi 2799 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
5 | df-sn 4623 | . . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
6 | 4, 5 | eqtr4di 2789 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
7 | 6 | inteqd 4948 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦}) |
8 | iotaval 6503 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
9 | 3, 7, 8 | 3eqtr4a 2797 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) |
10 | 9 | exlimiv 1933 | . 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 1539 = wceq 1541 ∃wex 1781 ∃!weu 2561 {cab 2708 {csn 4622 ∩ cint 4943 ℩cio 6482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-v 3475 df-un 3949 df-in 3951 df-ss 3961 df-sn 4623 df-pr 4625 df-uni 4902 df-int 4944 df-iota 6484 |
This theorem is referenced by: (None) |
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