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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1149 | 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 |
---|---|
bnj1149.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
bnj1149.2 | ⊢ (𝜑 → 𝐵 ∈ V) |
Ref | Expression |
---|---|
bnj1149 | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1149.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | bnj1149.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | unexg 7461 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 df-uni 4831 |
This theorem is referenced by: bnj1136 32166 bnj1413 32204 bnj1452 32221 bnj1489 32225 |
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