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Mirrors > Home > MPE Home > Th. List > inf5 | Structured version Visualization version GIF version |
Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 8894). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.) |
Ref | Expression |
---|---|
inf5 | ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 8900 | . 2 ⊢ ω ∈ V | |
2 | infeq5i 8893 | . 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1742 ∈ wcel 2050 Vcvv 3416 ⊊ wpss 3831 ∪ cuni 4712 ωcom 7396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pr 5186 ax-un 7279 ax-inf2 8898 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-tr 5031 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-om 7397 |
This theorem is referenced by: (None) |
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