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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 7448 | . . 3 ⊢ 2𝑜 = suc 1𝑜 | |
2 | 2on 7455 | . . . 4 ⊢ 2𝑜 ∈ On | |
3 | 2on0 7456 | . . . 4 ⊢ 2𝑜 ≠ ∅ | |
4 | 2onn 7607 | . . . . 5 ⊢ 2𝑜 ∈ ω | |
5 | nnlim 6970 | . . . . 5 ⊢ (2𝑜 ∈ ω → ¬ Lim 2𝑜) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ¬ Lim 2𝑜 |
7 | onsucuni3 32391 | . . . 4 ⊢ ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → 2𝑜 = suc ∪ 2𝑜) | |
8 | 2, 3, 6, 7 | mp3an 1416 | . . 3 ⊢ 2𝑜 = suc ∪ 2𝑜 |
9 | 1, 8 | eqtr3i 2634 | . 2 ⊢ suc 1𝑜 = suc ∪ 2𝑜 |
10 | suc11reg 8399 | . 2 ⊢ (suc 1𝑜 = suc ∪ 2𝑜 ↔ 1𝑜 = ∪ 2𝑜) | |
11 | 9, 10 | mpbi 219 | 1 ⊢ 1𝑜 = ∪ 2𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ∪ cuni 4372 Oncon0 5640 Lim wlim 5641 suc csuc 5642 ωcom 6957 1𝑜c1o 7440 2𝑜c2o 7441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 df-1o 7447 df-2o 7448 |
This theorem is referenced by: finxpreclem4 32407 |
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