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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 2390. See nfiung 4953 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 4923 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiun.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiun.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2973 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfrex 3311 | . . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2986 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2977 | 1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 {cab 2801 Ⅎwnfc 2963 ∃wrex 3141 ∪ ciun 4921 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-iun 4923 |
This theorem is referenced by: iunab 4977 disjxiun 5065 ovoliunnul 24110 iundisjf 30341 iundisj2f 30342 iundisjfi 30521 iundisj2fi 30522 bnj1498 32335 trpredlem1 33068 trpredrec 33079 ss2iundf 40011 nfcoll 40599 fnlimcnv 41955 fnlimfvre 41962 fnlimabslt 41967 smfaddlem1 43046 smflimlem6 43059 smflim 43060 smfmullem4 43076 smflim2 43087 smflimsup 43109 smfliminf 43112 |
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