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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 36710 | . 2 ⊢ (⊤ → 𝑋 ≠ 𝑌) |
6 | 5 | mptru 1547 | 1 ⊢ 𝑋 ≠ 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2105 ≠ wne 2939 Oncon0 6364 suc csuc 6366 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 df-suc 6370 |
This theorem is referenced by: finxpreclem3 36738 |
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