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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 6165 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | iuneq1 4897 | . . 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 4979 | . 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 4976 | . . 3 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
8 | 7 | uneq2i 4087 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝐴}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
9 | 3, 4, 8 | 3eqtri 2825 | 1 ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 {csn 4525 ∪ ciun 4881 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-iun 4883 df-suc 6165 |
This theorem is referenced by: pwsdompw 9615 |
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