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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 1913 | . . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) | |
| 2 | bnj1146.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
| 3 | 2 | nf5i 2145 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | 
| 4 | nfv 1913 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵 | |
| 5 | 3, 4 | nfan 1898 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) | 
| 6 | eleq1w 2823 | . . . . . . 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 2343 | . . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | 
| 9 | df-rex 3070 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
| 10 | df-rex 3070 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) | |
| 11 | 8, 9, 10 | 3bitr4i 303 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵) | 
| 12 | 11 | abbii 2808 | . . 3 ⊢ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} | 
| 13 | df-iun 4992 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} | |
| 14 | df-iun 4992 | . . 3 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵} | |
| 15 | 12, 13, 14 | 3eqtr4i 2774 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐵 | 
| 16 | bnj1143 34805 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 ⊆ 𝐵 | |
| 17 | 15, 16 | eqsstri 4029 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∃wrex 3069 ⊆ wss 3950 ∪ ciun 4990 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-ss 3967 df-nul 4333 df-iun 4992 | 
| This theorem is referenced by: bnj1145 35008 | 
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