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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 34528 | . 2 ⊢ (⊤ → 𝑋 ≠ 𝑌) |
6 | 5 | mptru 1535 | 1 ⊢ 𝑋 ≠ 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ≠ wne 3013 Oncon0 6184 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-suc 6190 |
This theorem is referenced by: finxpreclem3 34556 |
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