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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 6271 | . . . . 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 2918 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
6 | suceloni 7530 | . . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On) | |
7 | 1, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On) |
8 | 4, 7 | eqeltrd 2915 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On) |
9 | eloni 6203 | . . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord 𝑋) |
11 | ordirr 6211 | . . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋) |
13 | eleq1 2902 | . . . . . 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 3025 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Ord word 6192 Oncon0 6193 suc csuc 6195 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 df-suc 6199 |
This theorem is referenced by: sucneqoni 34649 |
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