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Mirrors > Home > MPE Home > Th. List > nnwos | Structured version Visualization version GIF version |
Description: Well-ordering principle: any nonempty set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) |
Ref | Expression |
---|---|
nnwos.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
nnwos | ⊢ (∃𝑥 ∈ ℕ 𝜑 → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfrab1 3425 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ ℕ ∣ 𝜑} | |
2 | nfcv 2904 | . . 3 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑} | |
3 | 1, 2 | nnwof 12847 | . 2 ⊢ (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ {𝑥 ∈ ℕ ∣ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦) |
4 | ssrab2 4041 | . . . 4 ⊢ {𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ | |
5 | 4 | biantrur 532 | . . 3 ⊢ ({𝑥 ∈ ℕ ∣ 𝜑} ≠ ∅ ↔ ({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ {𝑥 ∈ ℕ ∣ 𝜑} ≠ ∅)) |
6 | rabn0 4349 | . . 3 ⊢ ({𝑥 ∈ ℕ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ ℕ 𝜑) | |
7 | 5, 6 | bitr3i 277 | . 2 ⊢ (({𝑥 ∈ ℕ ∣ 𝜑} ⊆ ℕ ∧ {𝑥 ∈ ℕ ∣ 𝜑} ≠ ∅) ↔ ∃𝑥 ∈ ℕ 𝜑) |
8 | df-rex 3071 | . . 3 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦)) | |
9 | rabid 3426 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑥 ∈ ℕ ∧ 𝜑)) | |
10 | df-ral 3062 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∀𝑦(𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦)) | |
11 | nnwos.1 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
12 | 11 | elrab 3649 | . . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜓)) |
13 | 12 | imbi1i 350 | . . . . . . . 8 ⊢ ((𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦) ↔ ((𝑦 ∈ ℕ ∧ 𝜓) → 𝑥 ≤ 𝑦)) |
14 | impexp 452 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℕ ∧ 𝜓) → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) | |
15 | 13, 14 | bitri 275 | . . . . . . 7 ⊢ ((𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) |
16 | 15 | albii 1822 | . . . . . 6 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) |
17 | 10, 16 | bitri 275 | . . . . 5 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) |
18 | 9, 17 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦) ↔ ((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))) |
19 | 18 | exbii 1851 | . . 3 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦) ↔ ∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))) |
20 | df-ral 3062 | . . . . . . 7 ⊢ (∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) | |
21 | 20 | anbi2i 624 | . . . . . 6 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ ((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦)))) |
22 | anass 470 | . . . . . 6 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ (𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))) | |
23 | 21, 22 | bitr3i 277 | . . . . 5 ⊢ (((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ (𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))) |
24 | 23 | exbii 1851 | . . . 4 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))) |
25 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)) ↔ ∃𝑥(𝑥 ∈ ℕ ∧ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))) | |
26 | 24, 25 | bitr4i 278 | . . 3 ⊢ (∃𝑥((𝑥 ∈ ℕ ∧ 𝜑) ∧ ∀𝑦(𝑦 ∈ ℕ → (𝜓 → 𝑥 ≤ 𝑦))) ↔ ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))) |
27 | 8, 19, 26 | 3bitri 297 | . 2 ⊢ (∃𝑥 ∈ {𝑥 ∈ ℕ ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑}𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))) |
28 | 3, 7, 27 | 3imtr3i 291 | 1 ⊢ (∃𝑥 ∈ ℕ 𝜑 → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃wex 1782 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3406 ⊆ wss 3914 ∅c0 4286 class class class wbr 5109 ≤ cle 11198 ℕcn 12161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 |
This theorem is referenced by: indstr 12849 infpnlem2 16791 |
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