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Mirrors > Home > MPE Home > Th. List > uniabio | Structured version Visualization version GIF version |
Description: Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
uniabio | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi 2737 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | 1 | biimpi 206 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) |
3 | df-sn 4178 | . . . 4 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
4 | 2, 3 | syl6eqr 2674 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
5 | 4 | unieqd 4446 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦}) |
6 | vex 3203 | . . 3 ⊢ 𝑦 ∈ V | |
7 | 6 | unisn 4451 | . 2 ⊢ ∪ {𝑦} = 𝑦 |
8 | 5, 7 | syl6eq 2672 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 {cab 2608 {csn 4177 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 df-uni 4437 |
This theorem is referenced by: iotaval 5862 iotauni 5863 |
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