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Mirrors > Home > MPE Home > Th. List > infunabs | Structured version Visualization version GIF version |
Description: An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
infunabs | ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ dom card) | |
2 | reldom 8515 | . . . . . 6 ⊢ Rel ≼ | |
3 | 2 | brrelex1i 5608 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V) |
4 | 3 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ V) |
5 | undjudom 9593 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | |
6 | 1, 4, 5 | syl2anc 586 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
7 | infdjuabs 9628 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ 𝐴) | |
8 | domentr 8568 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≼ 𝐴) | |
9 | 6, 7, 8 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ 𝐴) |
10 | unexg 7472 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
11 | 1, 4, 10 | syl2anc 586 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ∈ V) |
12 | ssun1 4148 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
13 | ssdomg 8555 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ⊆ (𝐴 ∪ 𝐵) → 𝐴 ≼ (𝐴 ∪ 𝐵))) | |
14 | 11, 12, 13 | mpisyl 21 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≼ (𝐴 ∪ 𝐵)) |
15 | sbth 8637 | . 2 ⊢ (((𝐴 ∪ 𝐵) ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 ∪ 𝐵)) → (𝐴 ∪ 𝐵) ≈ 𝐴) | |
16 | 9, 14, 15 | syl2anc 586 | 1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ωcom 7580 ≈ cen 8506 ≼ cdom 8507 ⊔ cdju 9327 cardccrd 9364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-oi 8974 df-dju 9330 df-card 9368 |
This theorem is referenced by: infunsdom1 9635 infxp 9637 |
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