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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 4670 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
2 | cardon 9838 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
3 | 2 | onordi 6425 | . . . . 5 ⊢ Ord (card‘𝐴) |
4 | orddisj 6353 | . . . . 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 10441 | . . . . . 6 ⊢ (card‘𝐴) ≈ 𝐴 |
8 | 7 | ensymi 8902 | . . . . 5 ⊢ 𝐴 ≈ (card‘𝐴) |
9 | unsnen.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
10 | fvex 6852 | . . . . . 6 ⊢ (card‘𝐴) ∈ V | |
11 | en2sn 8943 | . . . . . 6 ⊢ ((𝐵 ∈ V ∧ (card‘𝐴) ∈ V) → {𝐵} ≈ {(card‘𝐴)}) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ {𝐵} ≈ {(card‘𝐴)} |
13 | unen 8948 | . . . . 5 ⊢ (((𝐴 ≈ (card‘𝐴) ∧ {𝐵} ≈ {(card‘𝐴)}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) | |
14 | 8, 12, 13 | mpanl12 700 | . . . 4 ⊢ (((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
15 | 5, 14 | mpan2 689 | . . 3 ⊢ ((𝐴 ∩ {𝐵}) = ∅ → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
16 | 1, 15 | sylbir 234 | . 2 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)})) |
17 | df-suc 6321 | . 2 ⊢ suc (card‘𝐴) = ((card‘𝐴) ∪ {(card‘𝐴)}) | |
18 | 16, 17 | breqtrrdi 5145 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cun 3906 ∩ cin 3907 ∅c0 4280 {csn 4584 class class class wbr 5103 Ord word 6314 suc csuc 6317 ‘cfv 6493 ≈ cen 8838 cardccrd 9829 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-ac2 10357 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-er 8606 df-en 8842 df-card 9833 df-ac 10010 |
This theorem is referenced by: (None) |
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