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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 2374. See nfiung 5051 for a less restrictive version requiring more axioms. (Revised by GG, 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 5021 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiun.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiun.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2895 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfrexw 3314 | . . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2910 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2103 {cab 2711 Ⅎwnfc 2888 ∃wrex 3072 ∪ ciun 5019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-iun 5021 |
This theorem is referenced by: iunab 5077 disjxiun 5166 ttrclselem1 9790 ttrclselem2 9791 ovoliunnul 25554 iundisjf 32602 iundisj2f 32603 iundisjfi 32793 iundisj2fi 32794 bnj1498 35029 ss2iundf 43561 nfcoll 44165 fnlimcnv 45522 fnlimfvre 45529 fnlimabslt 45534 smfaddlem1 46618 smflimlem6 46631 smflim 46632 smfmullem4 46649 smflim2 46661 smflimsup 46683 smfliminf 46686 fsupdm 46697 finfdm 46701 |
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