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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 8699). 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 8705 | . 2 ⊢ ω ∈ V | |
2 | infeq5i 8698 | . 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 1852 ∈ wcel 2145 Vcvv 3351 ⊊ wpss 3725 ∪ cuni 4575 ωcom 7213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7097 ax-inf2 8703 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3589 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-tr 4888 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-om 7214 |
This theorem is referenced by: (None) |
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