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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-prexg | Structured version Visualization version GIF version | ||
| Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 37034 and ax-bj-bun 37038. Contrary to bj-prex 37041, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-prexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-pr 4629 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 2 | bj-snexg 37035 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
| 3 | bj-snexg 37035 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V) | |
| 4 | bj-unexg 37039 | . . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V) | |
| 5 | 2, 3, 4 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ∈ V) | 
| 6 | 1, 5 | eqeltrid 2845 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-bj-sn 37034 ax-bj-bun 37038 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: (None) | 
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