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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 8086 | . . 3 ⊢ 2o = suc 1o | |
2 | 2on 8094 | . . . 4 ⊢ 2o ∈ On | |
3 | 2on0 8096 | . . . 4 ⊢ 2o ≠ ∅ | |
4 | 2onn 8249 | . . . . 5 ⊢ 2o ∈ ω | |
5 | nnlim 7573 | . . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ¬ Lim 2o |
7 | onsucuni3 34784 | . . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o) | |
8 | 2, 3, 6, 7 | mp3an 1458 | . . 3 ⊢ 2o = suc ∪ 2o |
9 | 1, 8 | eqtr3i 2823 | . 2 ⊢ suc 1o = suc ∪ 2o |
10 | 1on 8092 | . . 3 ⊢ 1o ∈ On | |
11 | onuni 7488 | . . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On) | |
12 | 2, 11 | ax-mp 5 | . . 3 ⊢ ∪ 2o ∈ On |
13 | suc11 6262 | . . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)) | |
14 | 10, 12, 13 | mp2an 691 | . 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o) |
15 | 9, 14 | mpbi 233 | 1 ⊢ 1o = ∪ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ∪ cuni 4800 Oncon0 6159 Lim wlim 6160 suc csuc 6161 ωcom 7560 1oc1o 8078 2oc2o 8079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-om 7561 df-1o 8085 df-2o 8086 |
This theorem is referenced by: finxpreclem4 34811 |
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