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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 6396 | . . . . 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 2841 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
6 | onsuc 7738 | . . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On) | |
7 | 1, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On) |
8 | 4, 7 | eqeltrd 2838 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On) |
9 | eloni 6325 | . . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord 𝑋) |
11 | ordirr 6333 | . . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋) |
13 | eleq1 2825 | . . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋)) | |
14 | 13 | biimprd 247 | . . . . 5 ⊢ (𝑋 = 𝑌 → (𝑌 ∈ 𝑋 → 𝑋 ∈ 𝑋)) |
15 | 14 | con3d 152 | . . . 4 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 ∈ 𝑋 → ¬ 𝑌 ∈ 𝑋)) |
16 | 12, 15 | syl5com 31 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌 ∈ 𝑋)) |
17 | 5, 16 | mt2d 136 | . 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌) |
18 | 17 | neqned 2948 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 Ord word 6314 Oncon0 6315 suc csuc 6317 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6318 df-on 6319 df-suc 6321 |
This theorem is referenced by: sucneqoni 35775 |
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