![]() |
Mathbox for ML |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sucneqoni | 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 |
---|---|
sucneqoni.1 | ⊢ 𝑋 = suc 𝑌 |
sucneqoni.2 | ⊢ 𝑌 ∈ On |
Ref | Expression |
---|---|
sucneqoni | ⊢ 𝑋 ≠ 𝑌 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucneqoni.1 | . . . 4 ⊢ 𝑋 = suc 𝑌 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝑋 = suc 𝑌) |
3 | sucneqoni.2 | . . . 4 ⊢ 𝑌 ∈ On | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝑌 ∈ On) |
5 | 2, 4 | sucneqond 37280 | . 2 ⊢ (⊤ → 𝑋 ≠ 𝑌) |
6 | 5 | mptru 1544 | 1 ⊢ 𝑋 ≠ 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2103 ≠ wne 2942 Oncon0 6394 suc csuc 6396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-tr 5287 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-ord 6397 df-on 6398 df-suc 6400 |
This theorem is referenced by: finxpreclem3 37308 |
Copyright terms: Public domain | W3C validator |