Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iunsuc | Structured version Visualization version GIF version |
Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
iunsuc.1 | ⊢ 𝐴 ∈ V |
iunsuc.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunsuc | ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6179 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | iuneq1 4898 | . . 3 ⊢ (suc 𝐴 = (𝐴 ∪ {𝐴}) → ∪ 𝑥 ∈ suc 𝐴𝐵 = ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵 |
4 | iunxun 4980 | . 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝐴})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) | |
5 | iunsuc.1 | . . . 4 ⊢ 𝐴 ∈ V | |
6 | iunsuc.2 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
7 | 5, 6 | iunxsn 4977 | . . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
8 | 7 | uneq2i 4051 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
9 | 3, 4, 8 | 3eqtri 2765 | 1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 Vcvv 3398 ∪ cun 3842 {csn 4517 ∪ ciun 4882 suc csuc 6175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-v 3400 df-un 3849 df-in 3851 df-ss 3861 df-sn 4518 df-iun 4884 df-suc 6179 |
This theorem is referenced by: pwsdompw 9705 |
Copyright terms: Public domain | W3C validator |