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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 9654). 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 9660 | . 2 ⊢ ω ∈ V | |
2 | infeq5i 9653 | . 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 1774 ∈ wcel 2099 Vcvv 3470 ⊊ wpss 3946 ∪ cuni 4903 ωcom 7864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 ax-inf2 9658 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-om 7865 |
This theorem is referenced by: (None) |
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