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Mathbox for Jonathan Ben-Naim |
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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 3064 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
3 | iunss 5048 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | mpbir 230 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∀wral 3060 ⊆ wss 3948 ∪ ciun 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-v 3475 df-in 3955 df-ss 3965 df-iun 4999 |
This theorem is referenced by: bnj229 34359 bnj1128 34465 bnj1145 34468 |
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