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Mirrors > Home > MPE Home > Th. List > unsnen | Structured version Visualization version GIF version |
Description: Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
unsnen.1 | ⊢ 𝐴 ∈ V |
unsnen.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
unsnen | ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn 4607 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
2 | cardon 9357 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
3 | 2 | onordi 6263 | . . . . 5 ⊢ Ord (card‘𝐴) |
4 | orddisj 6197 | . . . . 5 ⊢ (Ord (card‘𝐴) → ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅ |
6 | unsnen.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
7 | 6 | cardid 9958 | . . . . . 6 ⊢ (card‘𝐴) ≈ 𝐴 |
8 | 7 | ensymi 8542 | . . . . 5 ⊢ 𝐴 ≈ (card‘𝐴) |
9 | unsnen.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
10 | fvex 6658 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
11 | en2sn 8576 | . . . . . 6 ⊢ ((𝐵 ∈ V ∧ (card‘𝐴) ∈ V) → {𝐵} ≈ {(card‘𝐴)}) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 ⊢ {𝐵} ≈ {(card‘𝐴)} |
13 | unen 8579 | . . . . 5 ⊢ (((𝐴 ≈ (card‘𝐴) ∧ {𝐵} ≈ {(card‘𝐴)}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) | |
14 | 8, 12, 13 | mpanl12 701 | . . . 4 ⊢ (((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
15 | 5, 14 | mpan2 690 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
16 | 1, 15 | sylbir 238 | . 2 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
17 | df-suc 6165 | . 2 ⊢ suc (card‘𝐴) = ((card‘𝐴) ∪ {(card‘𝐴)}) | |
18 | 16, 17 | breqtrrdi 5072 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 class class class wbr 5030 Ord word 6158 suc csuc 6161 ‘cfv 6324 ≈ cen 8489 cardccrd 9348 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-ac2 9874 |
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-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 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-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-wrecs 7930 df-recs 7991 df-1o 8085 df-er 8272 df-en 8493 df-card 9352 df-ac 9527 |
This theorem is referenced by: (None) |
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