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Mirrors > Home > MPE Home > Th. List > bnd | Structured version Visualization version GIF version |
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 7725), derived from the Collection Principle cp 9504. Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.) |
Ref | Expression |
---|---|
bnd | ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cp 9504 | . . 3 ⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) | |
2 | ralim 3082 | . . 3 ⊢ (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)) | |
3 | 1, 2 | eximii 1844 | . 2 ⊢ ∃𝑤(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) |
4 | 3 | 19.37iv 1957 | 1 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1787 ∀wral 3058 ∃wrex 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-reg 9205 ax-inf2 9253 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-om 7642 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-r1 9377 df-rank 9378 |
This theorem is referenced by: bnd2 9506 |
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