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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 8125 | . . 3 ⊢ 2o = suc 1o | |
2 | 2on 8132 | . . . 4 ⊢ 2o ∈ On | |
3 | 2on0 8133 | . . . 4 ⊢ 2o ≠ ∅ | |
4 | 2onn 8290 | . . . . 5 ⊢ 2o ∈ ω | |
5 | nnlim 7606 | . . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ¬ Lim 2o |
7 | onsucuni3 35150 | . . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o) | |
8 | 2, 3, 6, 7 | mp3an 1462 | . . 3 ⊢ 2o = suc ∪ 2o |
9 | 1, 8 | eqtr3i 2763 | . 2 ⊢ suc 1o = suc ∪ 2o |
10 | 1on 8131 | . . 3 ⊢ 1o ∈ On | |
11 | onuni 7521 | . . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On) | |
12 | 2, 11 | ax-mp 5 | . . 3 ⊢ ∪ 2o ∈ On |
13 | suc11 6269 | . . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)) | |
14 | 10, 12, 13 | mp2an 692 | . 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 1542 ∈ wcel 2113 ≠ wne 2934 ∅c0 4209 ∪ cuni 4793 Oncon0 6166 Lim wlim 6167 suc csuc 6168 ωcom 7593 1oc1o 8117 2oc2o 8118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-tr 5134 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-om 7594 df-1o 8124 df-2o 8125 |
This theorem is referenced by: finxpreclem4 35177 |
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