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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 1909 | . . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) | |
2 | bnj1146.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | nf5i 2134 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfv 1909 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵 | |
5 | 3, 4 | nfan 1894 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) |
6 | eleq1w 2808 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
7 | 6 | anbi1d 629 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) |
8 | 1, 5, 7 | cbvexv1 2332 | . . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) |
9 | df-rex 3060 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
10 | df-rex 3060 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
11 | 8, 9, 10 | 3bitr4i 302 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵) |
12 | 11 | abbii 2795 | . . 3 ⊢ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} |
13 | df-iun 4999 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} | |
14 | df-iun 4999 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} | |
15 | 12, 13, 14 | 3eqtr4i 2763 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐵 |
16 | bnj1143 34552 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 ⊆ 𝐵 | |
17 | 15, 16 | eqsstri 4011 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 {cab 2702 ∃wrex 3059 ⊆ wss 3944 ∪ ciun 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-v 3463 df-dif 3947 df-ss 3961 df-nul 4323 df-iun 4999 |
This theorem is referenced by: bnj1145 34755 |
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