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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 | abbi1 2884 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | df-sn 4567 | . . . 4 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
4 | 3 | unieqd 4851 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦}) |
5 | vex 3497 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | unisn 4857 | . 2 ⊢ ∪ {𝑦} = 𝑦 |
7 | 4, 6 | syl6eq 2872 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 {cab 2799 {csn 4566 ∪ cuni 4837 |
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-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-sn 4567 df-pr 4569 df-uni 4838 |
This theorem is referenced by: iotaval 6328 iotauni 6329 |
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