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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 37016 and ax-bj-bun 37020. Contrary to bj-prex 37023, this proof is intuitionistically valid and does not require ax-nul 5312. (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 4634 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | bj-snexg 37017 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
3 | bj-snexg 37017 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V) | |
4 | bj-unexg 37021 | . . 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 2843 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 {csn 4631 {cpr 4633 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-bj-sn 37016 ax-bj-bun 37020 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: (None) |
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