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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 9230). 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 9236 | . 2 ⊢ ω ∈ V | |
2 | infeq5i 9229 | . 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 1787 ∈ wcel 2112 Vcvv 3398 ⊊ wpss 3854 ∪ cuni 4805 ωcom 7622 |
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-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-om 7623 |
This theorem is referenced by: (None) |
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