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Mirrors > Home > MPE Home > Th. List > nnwof | Structured version Visualization version GIF version |
Description: Well-ordering principle: any nonempty set of positive integers has a least element. This version allows 𝑥 and 𝑦 to be present in 𝐴 as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nnwof.1 | ⊢ Ⅎ𝑥𝐴 |
nnwof.2 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
nnwof | ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnwo 12641 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣) | |
2 | nfcv 2907 | . . 3 ⊢ Ⅎ𝑤𝐴 | |
3 | nnwof.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑣 | |
5 | 3, 4 | nfralw 3150 | . . 3 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 |
6 | nfv 1917 | . . 3 ⊢ Ⅎ𝑤∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 | |
7 | breq1 5077 | . . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ≤ 𝑣 ↔ 𝑥 ≤ 𝑣)) | |
8 | 7 | ralbidv 3108 | . . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 ↔ ∀𝑣 ∈ 𝐴 𝑥 ≤ 𝑣)) |
9 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑣𝐴 | |
10 | nnwof.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
11 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑣 | |
12 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑣 𝑥 ≤ 𝑦 | |
13 | breq2 5078 | . . . . 5 ⊢ (𝑣 = 𝑦 → (𝑥 ≤ 𝑣 ↔ 𝑥 ≤ 𝑦)) | |
14 | 9, 10, 11, 12, 13 | cbvralfw 3366 | . . . 4 ⊢ (∀𝑣 ∈ 𝐴 𝑥 ≤ 𝑣 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
15 | 8, 14 | bitrdi 287 | . . 3 ⊢ (𝑤 = 𝑥 → (∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
16 | 2, 3, 5, 6, 15 | cbvrexfw 3368 | . 2 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑤 ≤ 𝑣 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
17 | 1, 16 | sylib 217 | 1 ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnfc 2887 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4257 class class class wbr 5074 ≤ cle 10998 ℕcn 11961 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-n0 12222 df-z 12308 df-uz 12571 |
This theorem is referenced by: nnwos 12643 |
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