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Mirrors > Home > MPE Home > Th. List > nfiun | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2381. See nfiung 4942 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
Ref | Expression |
---|---|
nfiun.1 | ⊢ Ⅎ𝑦𝐴 |
nfiun.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfiun | ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4912 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiun.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiun.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2968 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfrex 3306 | . . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2981 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 ∃wrex 3136 ∪ ciun 4910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-iun 4912 |
This theorem is referenced by: iunab 4966 disjxiun 5054 ovoliunnul 24035 iundisjf 30267 iundisj2f 30268 iundisjfi 30445 iundisj2fi 30446 bnj1498 32230 trpredlem1 32963 trpredrec 32974 ss2iundf 39882 nfcoll 40469 fnlimcnv 41824 fnlimfvre 41831 fnlimabslt 41836 smfaddlem1 42916 smflimlem6 42929 smflim 42930 smfmullem4 42946 smflim2 42957 smflimsup 42979 smfliminf 42982 |
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