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Mirrors > Home > MPE Home > Th. List > axinf2 | Structured version Visualization version GIF version |
Description: A standard version of
Axiom of Infinity, expanded to primitives, derived
from our version of Infinity ax-inf 8897 and Regularity ax-reg 8853.
This theorem should not be referenced in any proof. Instead, use ax-inf2 8900 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.) |
Ref | Expression |
---|---|
axinf2 | ⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7418 | . . 3 ⊢ ∅ ∈ ω | |
2 | peano2 7419 | . . . 4 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω) | |
3 | 2 | ax-gen 1758 | . . 3 ⊢ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω) |
4 | zfinf 8898 | . . . . . 6 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) | |
5 | 4 | inf2 8882 | . . . . 5 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) |
6 | 5 | inf3 8894 | . . . 4 ⊢ ω ∈ V |
7 | eleq2 2854 | . . . . 5 ⊢ (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω)) | |
8 | eleq2 2854 | . . . . . . 7 ⊢ (𝑥 = ω → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ω)) | |
9 | eleq2 2854 | . . . . . . 7 ⊢ (𝑥 = ω → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ ω)) | |
10 | 8, 9 | imbi12d 337 | . . . . . 6 ⊢ (𝑥 = ω → ((𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ ω → suc 𝑦 ∈ ω))) |
11 | 10 | albidv 1879 | . . . . 5 ⊢ (𝑥 = ω → (∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω))) |
12 | 7, 11 | anbi12d 621 | . . . 4 ⊢ (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) ↔ (∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)))) |
13 | 6, 12 | spcev 3525 | . . 3 ⊢ ((∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)) → ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))) |
14 | 1, 3, 13 | mp2an 679 | . 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) |
15 | 0el 4208 | . . . . 5 ⊢ (∅ ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ¬ 𝑧 ∈ 𝑦) | |
16 | df-rex 3094 | . . . . 5 ⊢ (∃𝑦 ∈ 𝑥 ∀𝑧 ¬ 𝑧 ∈ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)) | |
17 | 15, 16 | bitri 267 | . . . 4 ⊢ (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)) |
18 | sucel 6104 | . . . . . . 7 ⊢ (suc 𝑦 ∈ 𝑥 ↔ ∃𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))) | |
19 | df-rex 3094 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))) | |
20 | 18, 19 | bitri 267 | . . . . . 6 ⊢ (suc 𝑦 ∈ 𝑥 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))) |
21 | 20 | imbi2i 328 | . . . . 5 ⊢ ((𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) |
22 | 21 | albii 1782 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) |
23 | 17, 22 | anbi12i 617 | . . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) ↔ (∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))) |
24 | 23 | exbii 1810 | . 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) ↔ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))) |
25 | 14, 24 | mpbi 222 | 1 ⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 ∀wal 1505 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ∃wrex 3089 ∅c0 4180 suc csuc 6033 ωcom 7398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-reg 8853 ax-inf 8897 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 |
This theorem is referenced by: (None) |
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