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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 6237 | . . . . 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 2893 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
6 | suceloni 7508 | . . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On) | |
7 | 1, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On) |
8 | 4, 7 | eqeltrd 2890 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On) |
9 | eloni 6169 | . . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord 𝑋) |
11 | ordirr 6177 | . . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋) |
13 | eleq1 2877 | . . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋)) | |
14 | 13 | biimprd 251 | . . . . 5 ⊢ (𝑋 = 𝑌 → (𝑌 ∈ 𝑋 → 𝑋 ∈ 𝑋)) |
15 | 14 | con3d 155 | . . . 4 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 ∈ 𝑋 → ¬ 𝑌 ∈ 𝑋)) |
16 | 12, 15 | syl5com 31 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌 ∈ 𝑋)) |
17 | 5, 16 | mt2d 138 | . 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌) |
18 | 17 | neqned 2994 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Ord word 6158 Oncon0 6159 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-suc 6165 |
This theorem is referenced by: sucneqoni 34783 |
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