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Mirrors > Home > MPE Home > Th. List > Mathboxes > sucneqond | Structured version Visualization version GIF version |
Description: Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.) |
Ref | Expression |
---|---|
sucneqond.1 | ⊢ (𝜑 → 𝑋 = suc 𝑌) |
sucneqond.2 | ⊢ (𝜑 → 𝑌 ∈ On) |
Ref | Expression |
---|---|
sucneqond | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucneqond.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ On) | |
2 | sucidg 5803 | . . . . 5 ⊢ (𝑌 ∈ On → 𝑌 ∈ suc 𝑌) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ suc 𝑌) |
4 | sucneqond.1 | . . . 4 ⊢ (𝜑 → 𝑋 = suc 𝑌) | |
5 | 3, 4 | eleqtrrd 2704 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
6 | suceloni 7013 | . . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On) | |
7 | 1, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On) |
8 | 4, 7 | eqeltrd 2701 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On) |
9 | eloni 5733 | . . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord 𝑋) |
11 | ordirr 5741 | . . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋) |
13 | eleq1 2689 | . . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋)) | |
14 | 13 | biimprd 238 | . . . . 5 ⊢ (𝑋 = 𝑌 → (𝑌 ∈ 𝑋 → 𝑋 ∈ 𝑋)) |
15 | 14 | con3d 148 | . . . 4 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 ∈ 𝑋 → ¬ 𝑌 ∈ 𝑋)) |
16 | 12, 15 | syl5com 31 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌 ∈ 𝑋)) |
17 | 5, 16 | mt2d 131 | . 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌) |
18 | 17 | neqned 2801 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Ord word 5722 Oncon0 5723 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 |
This theorem is referenced by: sucneqoni 33214 |
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