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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 3066 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 |
3 | iunss 5046 | . 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 3062 ⊆ wss 3946 ∪ ciun 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-v 3477 df-in 3953 df-ss 3963 df-iun 4997 |
This theorem is referenced by: bnj229 33832 bnj1128 33938 bnj1145 33941 |
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