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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 6269 | . . . . 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 2916 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
6 | suceloni 7528 | . . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On) | |
7 | 1, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On) |
8 | 4, 7 | eqeltrd 2913 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On) |
9 | eloni 6201 | . . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord 𝑋) |
11 | ordirr 6209 | . . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋) |
13 | eleq1 2900 | . . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋)) | |
14 | 13 | biimprd 250 | . . . . 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 3023 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Ord word 6190 Oncon0 6191 suc csuc 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-suc 6197 |
This theorem is referenced by: sucneqoni 34650 |
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