![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2201 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | 1 | biimpi 118 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) |
3 | df-sn 3450 | . . . 4 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
4 | 2, 3 | syl6eqr 2138 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
5 | 4 | unieqd 3662 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦}) |
6 | vex 2622 | . . 3 ⊢ 𝑦 ∈ V | |
7 | 6 | unisn 3667 | . 2 ⊢ ∪ {𝑦} = 𝑦 |
8 | 5, 7 | syl6eq 2136 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1287 = wceq 1289 {cab 2074 {csn 3444 ∪ cuni 3651 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 df-uni 3652 |
This theorem is referenced by: iotaval 4986 iotauni 4987 |
Copyright terms: Public domain | W3C validator |