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Mirrors > Home > ILE Home > Th. List > uniabio | 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 2303 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | 1 | biimpi 120 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) |
3 | df-sn 3613 | . . . 4 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
4 | 2, 3 | eqtr4di 2240 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
5 | 4 | unieqd 3835 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦}) |
6 | vex 2755 | . . 3 ⊢ 𝑦 ∈ V | |
7 | 6 | unisn 3840 | . 2 ⊢ ∪ {𝑦} = 𝑦 |
8 | 5, 7 | eqtrdi 2238 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 = wceq 1364 {cab 2175 {csn 3607 ∪ cuni 3824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-uni 3825 |
This theorem is referenced by: iotaval 5207 iotauni 5208 |
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