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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 9629). 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 9635 | . 2 ⊢ ω ∈ V | |
2 | infeq5i 9628 | . 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 1773 ∈ wcel 2098 Vcvv 3466 ⊊ wpss 3942 ∪ cuni 4900 ωcom 7849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-om 7850 |
This theorem is referenced by: (None) |
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