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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1oequni2o | Structured version Visualization version GIF version |
Description: The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
1oequni2o | ⊢ 1o = ∪ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 8092 | . . 3 ⊢ 2o = suc 1o | |
2 | 2on 8100 | . . . 4 ⊢ 2o ∈ On | |
3 | 2on0 8102 | . . . 4 ⊢ 2o ≠ ∅ | |
4 | 2onn 8255 | . . . . 5 ⊢ 2o ∈ ω | |
5 | nnlim 7582 | . . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ¬ Lim 2o |
7 | onsucuni3 34530 | . . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o) | |
8 | 2, 3, 6, 7 | mp3an 1452 | . . 3 ⊢ 2o = suc ∪ 2o |
9 | 1, 8 | eqtr3i 2843 | . 2 ⊢ suc 1o = suc ∪ 2o |
10 | 1on 8098 | . . 3 ⊢ 1o ∈ On | |
11 | onuni 7497 | . . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On) | |
12 | 2, 11 | ax-mp 5 | . . 3 ⊢ ∪ 2o ∈ On |
13 | suc11 6287 | . . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)) | |
14 | 10, 12, 13 | mp2an 688 | . 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o) |
15 | 9, 14 | mpbi 231 | 1 ⊢ 1o = ∪ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 ∪ cuni 4830 Oncon0 6184 Lim wlim 6185 suc csuc 6186 ωcom 7569 1oc1o 8084 2oc2o 8085 |
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-pw 4537 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-lim 6189 df-suc 6190 df-om 7570 df-1o 8091 df-2o 8092 |
This theorem is referenced by: finxpreclem4 34557 |
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