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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 1911 | . . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) | |
2 | bnj1146.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | nf5i 2146 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfv 1911 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵 | |
5 | 3, 4 | nfan 1896 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) |
6 | eleq1w 2895 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
7 | 6 | anbi1d 631 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) |
8 | 1, 5, 7 | cbvexv1 2358 | . . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) |
9 | df-rex 3144 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
10 | df-rex 3144 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
11 | 8, 9, 10 | 3bitr4i 305 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵) |
12 | 11 | abbii 2886 | . . 3 ⊢ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} |
13 | df-iun 4913 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} | |
14 | df-iun 4913 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} | |
15 | 12, 13, 14 | 3eqtr4i 2854 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐵 |
16 | bnj1143 32057 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 ⊆ 𝐵 | |
17 | 15, 16 | eqsstri 4000 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 ∃wex 1776 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ⊆ wss 3935 ∪ ciun 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-iun 4913 |
This theorem is referenced by: bnj1145 32260 |
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