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Mirrors > Home > MPE Home > Th. List > Mathboxes > finorwe | Structured version Visualization version GIF version |
Description: If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.) |
Ref | Expression |
---|---|
finorwe | ⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ¬ ω ∈ V) | |
2 | soss 5617 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → ( < Or 𝐴 → < Or 𝑥)) | |
3 | 2 | com12 32 | . . . . . . . . 9 ⊢ ( < Or 𝐴 → (𝑥 ⊆ 𝐴 → < Or 𝑥)) |
4 | 3 | adantl 481 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → < Or 𝑥)) |
5 | vex 3482 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
6 | fineqv 9297 | . . . . . . . . . . 11 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
7 | 6 | biimpi 216 | . . . . . . . . . 10 ⊢ (¬ ω ∈ V → Fin = V) |
8 | 5, 7 | eleqtrrid 2846 | . . . . . . . . 9 ⊢ (¬ ω ∈ V → 𝑥 ∈ Fin) |
9 | wofi 9323 | . . . . . . . . . 10 ⊢ (( < Or 𝑥 ∧ 𝑥 ∈ Fin) → < We 𝑥) | |
10 | 9 | ancoms 458 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ < Or 𝑥) → < We 𝑥) |
11 | 8, 10 | sylan 580 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝑥) → < We 𝑥) |
12 | 1, 4, 11 | syl6an 684 | . . . . . . 7 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → < We 𝑥)) |
13 | ssid 4018 | . . . . . . . . 9 ⊢ 𝑥 ⊆ 𝑥 | |
14 | wereu 5685 | . . . . . . . . . . 11 ⊢ (( < We 𝑥 ∧ (𝑥 ∈ V ∧ 𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃!𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) | |
15 | reurex 3382 | . . . . . . . . . . 11 ⊢ (∃!𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦 → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) | |
16 | 14, 15 | syl 17 | . . . . . . . . . 10 ⊢ (( < We 𝑥 ∧ (𝑥 ∈ V ∧ 𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
17 | 5, 16 | mp3anr1 1457 | . . . . . . . . 9 ⊢ (( < We 𝑥 ∧ (𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
18 | 13, 17 | mpanr1 703 | . . . . . . . 8 ⊢ (( < We 𝑥 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
19 | 18 | ex 412 | . . . . . . 7 ⊢ ( < We 𝑥 → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
20 | 12, 19 | syl6 35 | . . . . . 6 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦))) |
21 | 20 | impd 410 | . . . . 5 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
22 | 21 | alrimiv 1925 | . . . 4 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
23 | df-fr 5641 | . . . 4 ⊢ ( < Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) | |
24 | 22, 23 | sylibr 234 | . . 3 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < Fr 𝐴) |
25 | simpr 484 | . . 3 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < Or 𝐴) | |
26 | df-we 5643 | . . 3 ⊢ ( < We 𝐴 ↔ ( < Fr 𝐴 ∧ < Or 𝐴)) | |
27 | 24, 25, 26 | sylanbrc 583 | . 2 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < We 𝐴) |
28 | 27 | ex 412 | 1 ⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 Or wor 5596 Fr wfr 5638 We wwe 5640 ωcom 7887 Fincfn 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 |
This theorem is referenced by: (None) |
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