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Mirrors > Home > MPE Home > Th. List > infeq5 | 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 (shown on the right-hand side in the form of omex 8709.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
infeq5 | ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pss 3727 | . . . . 5 ⊢ (𝑥 ⊊ ∪ 𝑥 ↔ (𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥)) | |
2 | unieq 4592 | . . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅) | |
3 | uni0 4613 | . . . . . . . . . 10 ⊢ ∪ ∅ = ∅ | |
4 | 2, 3 | syl6req 2807 | . . . . . . . . 9 ⊢ (𝑥 = ∅ → ∅ = ∪ 𝑥) |
5 | eqtr 2775 | . . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ ∅ = ∪ 𝑥) → 𝑥 = ∪ 𝑥) | |
6 | 4, 5 | mpdan 705 | . . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∪ 𝑥) |
7 | 6 | necon3i 2960 | . . . . . . 7 ⊢ (𝑥 ≠ ∪ 𝑥 → 𝑥 ≠ ∅) |
8 | 7 | anim1i 593 | . . . . . 6 ⊢ ((𝑥 ≠ ∪ 𝑥 ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)) |
9 | 8 | ancoms 468 | . . . . 5 ⊢ ((𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)) |
10 | 1, 9 | sylbi 207 | . . . 4 ⊢ (𝑥 ⊊ ∪ 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)) |
11 | 10 | eximi 1907 | . . 3 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)) |
12 | eqid 2756 | . . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
13 | eqid 2756 | . . . . 5 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) | |
14 | vex 3339 | . . . . 5 ⊢ 𝑥 ∈ V | |
15 | 12, 13, 14, 14 | inf3lem7 8700 | . . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V) |
16 | 15 | exlimiv 2003 | . . 3 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V) |
17 | 11, 16 | syl 17 | . 2 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ω ∈ V) |
18 | infeq5i 8702 | . 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) | |
19 | 17, 18 | impbii 199 | 1 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1628 ∃wex 1849 ∈ wcel 2135 ≠ wne 2928 {crab 3050 Vcvv 3336 ∩ cin 3710 ⊆ wss 3711 ⊊ wpss 3712 ∅c0 4054 ∪ cuni 4584 ↦ cmpt 4877 ↾ cres 5264 ωcom 7226 reccrdg 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-reg 8658 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-ral 3051 df-rex 3052 df-reu 3053 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-om 7227 df-wrecs 7572 df-recs 7633 df-rdg 7671 |
This theorem is referenced by: (None) |
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