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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1146 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1146.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Ref | Expression |
---|---|
bnj1146 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1874 | . . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) | |
2 | bnj1146.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | nf5i 2085 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfv 1874 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵 | |
5 | 3, 4 | nfan 1863 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) |
6 | eleq1w 2841 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
7 | 6 | anbi1d 621 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) |
8 | 1, 5, 7 | cbvexv1 2279 | . . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) |
9 | df-rex 3087 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
10 | df-rex 3087 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
11 | 8, 9, 10 | 3bitr4i 295 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵) |
12 | 11 | abbii 2837 | . . 3 ⊢ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} |
13 | df-iun 4790 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} | |
14 | df-iun 4790 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} | |
15 | 12, 13, 14 | 3eqtr4i 2805 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐵 |
16 | bnj1143 31742 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 ⊆ 𝐵 | |
17 | 15, 16 | eqsstri 3884 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∀wal 1506 ∃wex 1743 ∈ wcel 2051 {cab 2751 ∃wrex 3082 ⊆ wss 3822 ∪ ciun 4788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-v 3410 df-dif 3825 df-in 3829 df-ss 3836 df-nul 4173 df-iun 4790 |
This theorem is referenced by: bnj1145 31942 |
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