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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj226 | 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 |
---|---|
bnj226.1 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
bnj226 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj226.1 | . . 3 ⊢ 𝐵 ⊆ 𝐶 | |
2 | 1 | rgenw 3153 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
3 | iunss 4972 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | mpbir 233 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∀wral 3141 ⊆ wss 3939 ∪ ciun 4922 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-in 3946 df-ss 3955 df-iun 4924 |
This theorem is referenced by: bnj229 32160 bnj1128 32266 bnj1145 32269 |
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