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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 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
3 | iunss 4940 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | mpbir 234 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∀wral 3051 ⊆ wss 3853 ∪ ciun 4890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-v 3400 df-in 3860 df-ss 3870 df-iun 4892 |
This theorem is referenced by: bnj229 32531 bnj1128 32637 bnj1145 32640 |
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